Ab initio study of $\beta$-decay and pairing in $N=Z$ nuclei

Technical Summary: Ab Initio Study of $\beta$-Decay and Pairing in $N=Z$ Nuclei

Problem Statement
The paper addresses the need for a microscopic description of $\beta$-decay properties and Gamow-Teller (GT) strength distributions in waiting-point (WP) nuclei ($^{72}\text{Kr}$, $^{68}\text{Se}$, and $^{64}\text{Ge}$) relevant to the rapid proton capture (rp) process in astrophysical environments. Experimental data for these proton-rich nuclei are scarce due to measurement challenges near the proton drip line. Consequently, astrophysical models rely heavily on theoretical inputs. A specific focus is placed on understanding the role of neutron-proton ($np$) pairing correlations—specifically the interplay between isovector ($T=1$) and isoscalar ($T=0$) pairing—in the ground states of $N=Z$ nuclei and their impact on $\beta$-decay observables. The work also seeks to resolve the "quenching puzzle" regarding GT strengths without relying on empirical factors.

Methodology
The authors employ the ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) method.

• Hamiltonian: Calculations utilize the realistic nuclear force interaction EM1.8/2.0, derived from chiral effective field theory (EFT) including two-nucleon (2N) and three-nucleon (3N) forces.
• Framework: The VS-IMSRG decouples an effective Hamiltonian for the $fp$-shell valence space (outside a $^{40}\text{Ca}$ core) using continuous unitary transformations truncated at the two-body level (IMSRG(2)).
• Operators: GT operators are consistently evolved within the same framework to eliminate the need for phenomenological quenching factors. The parent nucleus is used as the reference state for operator evolution.
• Pairing Analysis: To isolate pairing effects, the authors construct schematic pairing operators for isovector ($P_{01}$) and isoscalar ($P_{10}$) channels. They subtract these terms from the VS-IMSRG interaction to create modified Hamiltonians ($VS\text{-}IMSRG\text{-}P_{01}$ and $VS\text{-}IMSRG\text{-}P_{10}$), allowing for a direct quantification of each pairing channel's contribution to observables.
• Calculations: Half-lives are computed by summing partial decay rates to $1^+$ states within the kinematically allowed $Q$-value window. GT strength distributions ($B(GT)$) are calculated using the Lanczos strength function method.

Key Results

1. Half-Lives and Decay Intensity:

   • The calculated half-lives for $^{72}\text{Kr}$ and $^{68}\text{Se}$ are consistent with experimental data and shell model results (KB3G), though the $^{64}\text{Ge}$ half-life is severely underestimated. The authors attribute this discrepancy to the IMSRG(2) truncation, which may not fully capture the spectrum of $1^+$ states.
   • Nearly the entire decay intensity (approx. 95%) is confined within an excitation energy of $\approx 1$ MeV in the daughter nuclei, with dominant branching to the first few $1^+$ states.

2. Pairing Condensates:

   • Isoscalar ($T=0$): The study finds no evidence of an isoscalar condensate in the ground states of the investigated $N=Z$ nuclei. The overlap between physical ground states and ideal isoscalar condensate states is less than 1%.
   • Isovector ($T=1$): While the ground states show some overlap with isovector condensates, the values are marginal (e.g., 10.1% for $^{72}\text{Kr}$, 4.3% for $^{68}\text{Se}$). The results do not support a dominant isovector pairing condensate-like phase in these waiting-point nuclei, unlike in lighter doubly magic systems like $^{56}\text{Ni}$ where overlaps are higher.

3. Impact on Ground State Energies and $B(GT)$:

   • Ground State Energies: Isovector pairing contributions to ground state energies are systematically larger than isoscalar contributions across the $N=Z$ isotopes studied.
   • $B(GT)$ Distributions: Pairing correlations significantly influence the accumulated $B(GT)$ strength within the low-energy $Q$-value window.
     • For $^{68}\text{Se}$ decay, omitting isoscalar pairing reduces the accumulated $B(GT)$ by $\approx 0.3$ units.
     • For $^{64}\text{Ge}$ and $^{72}\text{Kr}$, excluding either isoscalar or isovector pairing modifies the accumulated strength by approximately 20–40%.
   • High-Energy Behavior: In the full excitation spectrum, isovector pairing tends to suppress $B(GT)$ strengths at high excitation energies ($E_x \gtrsim 10$ MeV) and shifts states, while isoscalar pairing has a negligible impact at high energies.

Significance and Claims
The paper claims to provide a microscopic picture of $\beta$-decay strengths and pairing correlations in $N=Z$ nuclei far from the stability line, derived directly from fundamental chiral forces without empirical quenching.

• It establishes that while $np$ pairing is present, the ground states of these specific waiting-point nuclei do not exhibit a condensate-like phase for either isoscalar or isovector pairing.
• The work demonstrates that pairing correlations are crucial for shaping the low-energy $B(GT)$ distributions, which are critical inputs for astrophysical rp-process models.
• The authors note that while the VS-IMSRG framework is powerful, the underestimation of the $^{64}\text{Ge}$ half-life suggests that higher-order many-body approximations (e.g., IMSRG(3)) and consistent two-body currents (2BCs) are necessary for more reliable predictions in proton-rich systems.

The study concludes that the current VS-IMSRG approach, despite limitations in specific half-life predictions, offers a robust theoretical baseline for interpreting current measurements and guiding future experiments at rare-isotope beam facilities.