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Benchmarking projected generator coordinate method for nuclear Gamow-Teller transitions

R. N. Chen, X. Lian, J. M. Yao, C. L. Bai

TL;DR

This work tackles the challenge of describing Gamow-Teller transitions and the two-neutrino double-beta NMEs by extending the quantum-number projected generator coordinate method (PGCM) to include odd–odd intermediate states. It combines symmetry-projected Hartree-Fock-Bogoliubov states with a two-quasiparticle construction for odd–odd nuclei, solved via Hill-Wheeler-Griffin mixing, and benchmarks the approach against exact shell-model solutions and CI with particle-hole truncations using a GXPF1A-based $fp$-shell Hamiltonian. The results show that PGCM reproduces GT strength distributions in Ca and Ti isotopes reasonably well, with performance comparable to or slightly better than CI(2p2h) in many cases, though high-lying states reveal missing correlations. For the $2\nu\beta\beta$ decay of $^{48}$Ca, the PGCM yields an overall NME that overestimates the shell-model result by about 57%, primarily due to an overestimated initial GT matrix element; this highlights the need for further improvements such as expanding the generator-coordinate space and incorporating IMSRG-evolved operators. Overall, the work demonstrates the PGCM’s viability for beta-decay observables near closed shells and points to concrete avenues for systematic enhancements to improve predictive power for more complex nuclei.

Abstract

In this work, we aim to achieve a minimal extension of the quantum-number projected generator coordinate method (PGCM) to describe Gamow-Teller (GT) transition strengths in even-even nuclei and to compute the NME of $2νββ$ decay. Within the PGCM framework, the wave functions of odd-odd nuclei are constructed as superpositions of neutron and proton quasiparticle configurations built on quasiparticle vacua constrained to have, on average, odd neutron and odd proton particle numbers. The angular momentum and particle numbers associated with the underlying mean-field states are restored through projection techniques. Using a shell-model Hamiltonian defined in the $fp$ shell, we assess the validity of this approach by benchmarking GT transitions in calcium and titanium isotopes, as well as the $2νββ$ decay of $^{48}$Ca to $^{48}$Ti, against exact solutions. For comparison, we also confront our results with those obtained from configuration-interaction calculations employing different particle-hole truncation schemes, both with and without in-medium similarity renormalization group (IMSRG) evolution.

Benchmarking projected generator coordinate method for nuclear Gamow-Teller transitions

TL;DR

This work tackles the challenge of describing Gamow-Teller transitions and the two-neutrino double-beta NMEs by extending the quantum-number projected generator coordinate method (PGCM) to include odd–odd intermediate states. It combines symmetry-projected Hartree-Fock-Bogoliubov states with a two-quasiparticle construction for odd–odd nuclei, solved via Hill-Wheeler-Griffin mixing, and benchmarks the approach against exact shell-model solutions and CI with particle-hole truncations using a GXPF1A-based -shell Hamiltonian. The results show that PGCM reproduces GT strength distributions in Ca and Ti isotopes reasonably well, with performance comparable to or slightly better than CI(2p2h) in many cases, though high-lying states reveal missing correlations. For the decay of Ca, the PGCM yields an overall NME that overestimates the shell-model result by about 57%, primarily due to an overestimated initial GT matrix element; this highlights the need for further improvements such as expanding the generator-coordinate space and incorporating IMSRG-evolved operators. Overall, the work demonstrates the PGCM’s viability for beta-decay observables near closed shells and points to concrete avenues for systematic enhancements to improve predictive power for more complex nuclei.

Abstract

In this work, we aim to achieve a minimal extension of the quantum-number projected generator coordinate method (PGCM) to describe Gamow-Teller (GT) transition strengths in even-even nuclei and to compute the NME of decay. Within the PGCM framework, the wave functions of odd-odd nuclei are constructed as superpositions of neutron and proton quasiparticle configurations built on quasiparticle vacua constrained to have, on average, odd neutron and odd proton particle numbers. The angular momentum and particle numbers associated with the underlying mean-field states are restored through projection techniques. Using a shell-model Hamiltonian defined in the shell, we assess the validity of this approach by benchmarking GT transitions in calcium and titanium isotopes, as well as the decay of Ca to Ti, against exact solutions. For comparison, we also confront our results with those obtained from configuration-interaction calculations employing different particle-hole truncation schemes, both with and without in-medium similarity renormalization group (IMSRG) evolution.
Paper Structure (9 sections, 24 equations, 7 figures)

This paper contains 9 sections, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The methods
  3. The Hamiltonian
  4. The projected generator coordinate method
  5. The configuration-interaction (CI) method with particle-hole truncation
  6. Results and discussion
  7. The GT transition matrix elements
  8. The NME of the 2$\nu\beta\beta$ decay in $^{48}$Ca
  9. Summary

Figures (7)

  • Figure 1: (Color online) The distribution of ${\rm GT}^-$ transition strength $B({\rm GT}^-: 0^+_1 \to 1^+_m)$ from $^{42-48}$Ca to $^{42-48}$Sc as a function of the energy of the $1^+_m$ states in $^{42-48}$Sc isotopes from the PGCM calculation, in comparison with the results of exact shell-model and CI(2p2h) calculations.
  • Figure 2: (Color online) The same as Fig. \ref{['fig:Ca-isotopes']}, but for the ${\rm GT}^+$ transitions from $^{42-48}$Ti to $^{42-48}$Sc, respectively.
  • Figure 3: (Color online) (a) The distribution of ${\rm GT}^-$ transition strength for $^{44}$Ca as a function of the energy of the $1^+$ states in $^{44}$Sc. (b) The cumulated GT transition strength. The results of the PGCM calculations with the wave function of $^{44}$Ca constructed using either the pure spherical state or GCM state are compared to the shell-model results.
  • Figure 4: (Color online) The distribution of GT transition strength for (a) $^{48}$Ca and (b) $^{48}$Ti as a function of the energy of the $1^+$ states in $^{48}$Sc from different calculations.
  • Figure 5: (Color) The distributions of ${\rm GT}^-$ transition from $^{48}$Ca (a) and ${\rm GT}^+$ transition from $^{48}$Ti (b) as a function of the energy of the $1^+$ states in $^{48}$Sc, obtained from CI(1p1h), CI(2p2h) and shell-model calculations.
  • ...and 2 more figures