⚛️ nuclear theory

Evaluation of $2\nu\beta\beta$ decay nuclear matrix elements using Lanczos techniques preserving Gamow-Teller strengths from both parent and daughter nuclei

This paper introduces and validates two efficient Lanczos-based shell-model approaches, DLSF and BLSF, for calculating two-neutrino double-beta decay nuclear matrix elements that preserve Gamow-Teller strengths in both parent and daughter nuclei, successfully benchmarking them on $^{48}$ Ca and applying them to other candidates like $^{58}$ Ni, $^{76}$ Ge, and $^{130}$ Te.

Original authors: Deepak Patel, Anil Kumar, Noritaka Shimizu, Yutaka Utsuno

Published 2026-11-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Deepak Patel, Anil Kumar, Noritaka Shimizu, Yutaka Utsuno

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the heart of the atom, protons and neutrons are bound together in a delicate balance. Sometimes, this balance tips, and a nucleus seeks stability by transforming itself. In a rare and specific type of transformation called double-beta decay, two neutrons inside a nucleus simultaneously turn into two protons, ejecting two electrons and two invisible particles called neutrinos. This process is so slow that for some elements, it happens only once every few trillion years. While scientists have long observed this two-neutrino version of the decay, they are hunting for a ghostly, even rarer variation where no neutrinos are emitted at all. Finding that missing neutrinoless version would prove that neutrinos are their own antiparticles, a discovery that could explain why the universe is made of matter rather than being an empty void of antimatter. To find this rare event, physicists must first master the rules of the common, two-neutrino version, calculating exactly how the nucleus behaves during the transformation. This requires mapping the invisible energy states of the atom with extreme precision, a task that has long been a computational bottleneck for scientists trying to understand the fundamental building blocks of our world.

A team of researchers has now developed a new, highly efficient way to perform these calculations, bypassing the massive computational hurdles that have previously limited their accuracy. Instead of trying to list every single possible energy state the nucleus could pass through during the decay—a task that becomes impossible as the atom gets heavier—they used a mathematical shortcut to capture the essential behavior of the entire system. By focusing on the specific "doorway" states that initiate the transformation, they were able to reconstruct the full picture of the decay without needing to compute thousands of intermediate steps individually. This approach allowed them to preserve the total strength of the nuclear forces involved from both the starting and ending points of the decay, ensuring that no crucial information was lost in the simplification. The result is a method that is not only faster but also more reliable for the heavy atoms where the rare, neutrinoless decay might be hiding.

The researchers tested their new techniques on several candidate atoms, starting with calcium-48, a well-studied element often used as a benchmark. By expanding the mathematical "room" in which they calculated the atom's behavior to include more complex movements of its particles, they found that the predicted strength of the decay dropped slightly, bringing the theoretical number closer to what experiments have actually measured. This adjustment suggests that looking at the atom with a wider lens provides a more accurate description of its internal structure. They then applied their methods to nickel-58, germanium-76, and tellurium-130, covering a range of atomic weights. For nickel-58, this work represents the first time a detailed shell-model calculation has been performed for this specific type of decay, revealing a theoretical value that is far smaller than current experimental limits. This discrepancy indicates that while the theory is robust, the experiments need to become significantly more sensitive to detect the process, if it occurs at all.

To ensure their new shortcuts were not introducing errors, the team performed a rigorous check. They compared the sum of all the intermediate steps they calculated against a direct measurement of the total force between the starting and ending states, a value that can be determined independently. The two numbers matched perfectly across all the atoms they studied. This agreement confirms that their method successfully recovers the complete picture of the nuclear transformation, capturing every relevant contribution without needing to explicitly calculate every single intermediate state. The researchers found that their new approaches converged on the correct answer with far fewer computational steps than traditional methods, making it feasible to study heavier and more complex nuclei that were previously out of reach.

The implications of this work extend beyond just refining numbers for known decays. By providing a reliable and efficient way to calculate the nuclear matrix elements—the core numbers that connect theory to experiment—this study strengthens the foundation for the search for the neutrinoless decay. If the rare, neutrinoless version of this process exists, its rate depends entirely on these nuclear calculations. By demonstrating that their methods can accurately model the common decay while preserving the total strength of the nuclear forces, the researchers have provided a trusted tool for the global community. Their work suggests that the path forward lies in expanding the models to include more complex particle interactions, as seen in their improved results for calcium-48, and in pushing experimental detectors to greater sensitivities, as the case of nickel-58 clearly demands. The hunt for the universe's deepest secrets continues, but the map of the atomic landscape is now drawn with greater clarity and confidence.

Technical Summary: Evaluation of 2νββ Decay Nuclear Matrix Elements Using Lanczos Techniques

Problem Statement
The calculation of nuclear matrix elements (NMEs) for two-neutrino double-beta ( $2\nu\beta\beta$ ) decay and two-neutrino double-electron capture ( $2\nu$ ECEC) is essential for interpreting experimental data and providing benchmarks for nuclear structure models. These NMEs require a summation over the complete set of $1^+$ intermediate eigenstates of the daughter nucleus. However, in medium- and heavy-mass nuclei, the dimension of the Hamiltonian matrix grows rapidly, making the explicit calculation of all intermediate $1^+$ states computationally prohibitive. Conventional shell-model calculations often rely on truncated model spaces or limited numbers of intermediate states, potentially omitting a significant fraction of the Gamow-Teller (GT) strength and compromising the convergence and reliability of the NMEs.

Methodology
The authors propose two efficient shell-model approaches based on Lanczos techniques to evaluate $2\nu\beta\beta$ NMEs without explicitly calculating the complete set of intermediate $1^+$ eigenstates:

Dual-sided Lanczos Strength-Function (DLSF): This method constructs starting vectors by applying GT operators to both the initial and final $0^+$ ground states. It performs Lanczos iterations for both sides, generating two sets of approximate $1^+$ states. These sets are combined and orthogonalized using Löwdin orthogonalization, followed by a rediagonalization of the Hamiltonian in the resulting subspace to obtain Ritz vectors.
Block-Lanczos Strength-Function (BLSF): This method utilizes a block-Lanczos procedure where the initial vectors (derived from both parent and daughter nuclei) form a block. The Hamiltonian acts simultaneously on this block, generating a block Krylov subspace. The projected Hamiltonian takes a symmetric block-tridiagonal form, which is diagonalized to yield Ritz vectors.

Both methods are designed to preserve the total GT strengths associated with both the initial and final nuclei simultaneously. They are implemented within the KSHELL code. To validate the methods, the authors compare the energy-denominator-free sums of the NMEs against exact double-Gamow-Teller (DGT) matrix elements calculated directly between the initial and final ground states using closure relations.

Key Results
The methods were benchmarked and applied to several nuclei across different mass regions:

$^{48}$ Ca (fp shell): Using the GXPF1A interaction, both DLSF and BLSF converged to an NME of 0.0511 within 30 iterations, matching the result of exact diagonalization using 2,500 intermediate states. The methods successfully preserved GT strengths for both the $^{48}$ Ca $\to$ $^{48}$ Sc and $^{48}$ Ti $\to$ $^{48}$ Sc transitions, a feature not simultaneously achieved by the previous single-sided Lanczos (SLSF) method.
$^{48}$ Ca (sdpf shell): Extending the model space to include cross-shell excitations up to $2\hbar\omega$ using the SDPF-MU interaction yielded a converged NME of 0.0463. This value is closer to the experimental estimate (derived from half-life data) than the fp-shell result, suggesting that model space enlargement improves the description of the decay.
$^{58}$ Ni (fp shell): The authors present the first shell-model calculation of the $2\nu\beta^+EC/2\nu$ ECEC NME for $^{58}$ Ni. The converged NME is approximately 0.1250. When compared to the current experimental lower limit on the half-life, the theoretical NME is significantly smaller, indicating that current experiments lack the sensitivity to constrain the NME effectively.
$^{76}$ Ge and $^{130}$ Te: For $^{76}$ Ge (JUN45 interaction) and $^{130}$ Te (SN100PN interaction), the methods achieved full convergence (NMEs of 0.1656 and 0.0795, respectively) with relatively few iterations (75 and 60). In the case of $^{130}$ Te, exact diagonalization of up to 1,000 states showed slow convergence, highlighting the necessity of the Lanczos approaches for larger model spaces.
Validation: For all tested nuclei, the energy-denominator-free sums obtained via DLSF and BLSF agreed precisely with the exact DGT matrix elements calculated directly. This confirms that the methods correctly recover the complete intermediate-state contributions.

Significance and Claims
The paper claims that the DLSF and BLSF methods provide a robust and computationally efficient framework for calculating $2\nu\beta\beta$ NMEs in large shell-model spaces. By simultaneously preserving GT strengths from both the parent and daughter nuclei, these methods offer a more symmetric and consistent treatment of intermediate-state contributions compared to previous single-sided approaches. The successful reproduction of exact DGT matrix elements serves as an independent validation that the methods capture the full GT strength accessible within the adopted model spaces. The results for $^{48}$ Ca suggest that enlarging the model space to include cross-shell excitations yields NMEs closer to experimental values, lending support to the use of such enlarged spaces for predicting neutrinoless double-beta ( $0\nu\beta\beta$ ) NMEs. The study also underscores the need for more sensitive experimental measurements of the $2\nu\beta^+EC$ half-life in $^{58}$ Ni to provide meaningful constraints on theoretical models.

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