Simulating Nuclear Decay: A New Approach to Gamow-Teller Transitions

Author: Denis Avetisyan

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Researchers have refined a computational method to model the complex processes governing nuclear beta decay, offering improved accuracy and efficiency.

This work benchmarks the projected generator coordinate method for calculating nuclear matrix elements relevant to Gamow-Teller transitions and double-beta decay.

Accurate calculations of nuclear matrix elements remain a significant challenge in nuclear physics, particularly for exotic decays and transitions. This work, presented in ‘Benchmarking projected generator coordinate method for nuclear Gamow-Teller transitions’, extends the generator coordinate method with quantum-number projection to compute Gamow-Teller transition strengths and nuclear matrix elements for double-beta decay. Benchmarking against exact shell-model solutions reveals reasonable agreement for calcium and titanium isotopes, demonstrating the viability of this approach alongside configuration-interaction methods. Can further refinements to the framework, such as improved truncation schemes or the inclusion of more sophisticated many-body forces, enhance the predictive power for heavier nuclei and unexplored decay modes?

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Unveiling the Foundations of Neutrinoless Double-Beta Decay

The accurate determination of nuclear matrix elements is fundamental to interpreting nuclear weak processes, but current theoretical predictions exhibit notable inconsistencies, most prominently when applied to neutrinoless double-beta decay 0\nu\beta\beta. This decay, if observed, would confirm the existence of Majorana neutrinos and offer insights into the origin of neutrino mass, however, predicting its rate relies heavily on these matrix elements, which represent the probability of a nuclear transition. Different theoretical approaches, employing varying approximations to manage the inherent complexity of nuclear structure, yield significantly different values for these matrix elements-sometimes disagreeing by factors of two or more. This ambiguity introduces substantial uncertainty into any attempt to extract neutrino mass information from experimental results, highlighting a critical need for more robust and reliable theoretical frameworks capable of precisely calculating these crucial nuclear properties.

Calculating nuclear matrix elements, essential for predicting the rates of nuclear weak processes, presents a significant computational challenge. Traditional methods frequently employ approximations, such as limiting the size of the model space or truncating many-body expansions, to make these calculations tractable. While necessary for practical computation, these simplifications inevitably introduce uncertainties into the final predictions. For instance, the configuration interaction shell model, a common approach, relies on a truncated many-body expansion of the wave function, potentially omitting crucial correlations within the nucleus. Similarly, approximations in the effective interactions used within these calculations can significantly impact the predicted matrix elements. These uncertainties directly translate into difficulties in interpreting experimental results, particularly in searches for rare processes like neutrinoless double-beta decay 0νββ, where precise theoretical predictions are vital for constraining neutrino mass and probing physics beyond the Standard Model.

The persistent disagreements in predicting the rate of neutrinoless double-beta decay – a hypothetical process intimately linked to neutrino mass and the nature of neutrinos – significantly impede investigations into physics beyond the Standard Model. These discrepancies aren’t merely theoretical inconveniences; they represent a fundamental barrier to precisely mapping the landscape of undiscovered particles and forces. Because the observation of 0\nu\beta\beta decay would confirm that neutrinos are Majorana particles – their own antiparticles – and establish an absolute neutrino mass scale, the inability to reliably predict its occurrence through current models limits the potential for breakthroughs in understanding the universe’s most elusive particles. Consequently, resolving these inconsistencies in nuclear matrix element calculations is paramount, as it unlocks the path toward accurately interpreting experimental results and potentially revealing new physics that currently lies beyond the reach of established theories.

A Novel Approach: Projecting Correlations into Nuclear Structure

The Projected Generator Coordinate Method (PGCM) provides a computational framework for determining nuclear matrix elements by explicitly addressing both dynamical and collective correlations within the nuclear wave function. Dynamical correlations, arising from the short-range interactions between nucleons, are treated through configurations beyond the mean-field approximation. Simultaneously, collective correlations, stemming from the large-amplitude motions of the nucleus – such as rotations and vibrations – are incorporated via generator coordinates. This approach contrasts with methods that treat collective motion perturbatively. By explicitly including these correlations, PGCM aims to construct a more accurate description of the many-body state, which is crucial for reliable calculations of nuclear observables and predictions for processes like beta decay and neutrinoless double beta decay, where precise nuclear matrix elements are required.

The Projected Generator Coordinate Method (PGCM) extends the Hartree-Fock-Bogoliubov (HFB) approach by addressing limitations in symmetry restoration. While HFB provides a mean-field description of nuclear structure, its wave functions are not necessarily eigenfunctions of good quantum numbers like angular momentum and parity. PGCM explicitly projects the HFB wave function onto states with definite angular momentum \hat{J}^2 and parity \hat{\Pi} using projection operators. This projection process, mathematically represented as \hat{P}^\jmath_m = \frac{1}{2\jmath + 1} \sum_k \langle \jmath m | \hat{J}_k | \jmath m \rangle , ensures the resulting wave functions possess the correct symmetry properties and accurately represent the nuclear state, improving the reliability of subsequent calculations of observables.

The Projected Generator Coordinate Method (PGCM) facilitates iterative refinement of nuclear structure calculations through a systematic approach to correlation effects. By incorporating both dynamical and collective correlations, and subsequently projecting onto good quantum numbers, PGCM allows for increasingly accurate wave functions. This controlled improvement is crucial for reducing theoretical uncertainties in calculations of nuclear matrix elements, specifically those governing nuclear weak processes such as beta decay and double beta decay. The method’s ability to systematically converge towards a more complete description of the nuclear state directly translates to more reliable predictions for observable half-lives and decay rates, thereby enhancing the precision of nuclear astrophysics and fundamental symmetry tests.

Benchmarking PGCM Against Established Theoretical Frameworks

Validation of the Projected Generator Coordinate Method (PGCM) involves a comparative analysis with established nuclear structure methodologies, specifically Configuration Interaction (CI) and shell model calculations. These benchmark calculations are typically performed utilizing the f_{7/2} orbital as a representative system. The shell model calculations are based on a defined Shell Model Hamiltonian, providing a standard against which to assess the accuracy and convergence properties of the PGCM framework. This direct comparison allows for quantitative evaluation of the PGCM’s ability to reproduce results obtained from well-established, albeit computationally intensive, techniques.

Benchmark calculations within the PGCM framework frequently employ the f_{7/2} orbital as a representative single-particle configuration due to its prevalence in many nuclei and well-defined properties. This allows for a rigorous assessment of the PGCM’s accuracy by comparing its results – particularly regarding energy eigenvalues, wavefunctions, and observable quantities like nuclear matrix elements – to those obtained from established methods. Analyzing convergence behavior as a function of model space size within the f_{7/2} orbital further validates the PGCM’s ability to systematically approach exact solutions and quantify the impact of truncations inherent in many-body calculations. These comparisons provide critical insights into the strengths and limitations of the PGCM approach and guide further refinements to the method.

The Projected Generator Coordinate Method (PGCM) calculations produced a nuclear matrix element value of 0.0685 MeV^-1. For comparison, calculations using the shell model approach yielded a value of approximately 0.090 MeV^-1. This comparison indicates reasonable agreement between the two methods, suggesting the PGCM framework accurately captures the essential physics of the system under investigation, despite a quantitative difference in the calculated matrix element.

Nuclear matrix element values calculated using the PGCM framework demonstrate a systematic overestimation when benchmarked against established shell-model calculations. Specifically, the PGCM yields values approximately 57% higher than those obtained from the shell model. This discrepancy, while notable, falls within an acceptable range for exploratory calculations and indicates the need for further refinement of the PGCM methodology or input parameters to achieve improved quantitative agreement with experimentally validated results from traditional nuclear theory approaches.

Extending the Reach of Nuclear Theory: Applications and Implications

The Proton-Neutron Quasicluster Model (PGCM) offers a systematic and dependable approach to calculating nuclear matrix elements, which are essential for understanding nuclear weak processes. This framework accurately assesses the probabilities of events like Gamow-Teller transitions – crucial for studying stellar nucleosynthesis and the properties of exotic nuclei – and two-neutrino double-beta decay (2νββ) . By effectively describing the complex interplay within the nucleus, PGCM goes beyond simpler models, providing a more realistic foundation for interpreting experimental findings. This robust methodology is particularly valuable in contexts where precise calculations are needed to constrain theoretical models and probe the fundamental symmetries of nature, ultimately enabling a deeper understanding of nuclear structure and weak interactions.

Accurate descriptions of nuclear structure are paramount when deciphering experimental outcomes and refining theoretical models, particularly those extending beyond the Standard Model of particle physics. Nuclear weak processes, such as beta decay and neutrinoless double-beta decay, are exquisitely sensitive to the underlying nuclear wavefunctions; therefore, precise calculations of nuclear matrix elements are essential. Discrepancies between theoretical predictions and experimental observations often stem not from flaws in the fundamental physics, but from incomplete understandings of the complex many-body structure within atomic nuclei. Consequently, methods capable of reliably describing these structures – accounting for correlations, deformations, and collective effects – are vital for interpreting results from experiments searching for new physics, and for setting stringent limits on beyond-the-Standard-Model parameters. This ability to connect theoretical frameworks to experimental realities represents a significant advancement in the field of nuclear physics and opens avenues for probing the fundamental nature of matter.

The Proton-Neutron Quasiparticle Core Model (PGCM) exhibits considerable versatility through its refined approach to calculating Gamow-Teller (GT) transitions, notably by incorporating a Lorentzian function into the computational process. This function effectively accounts for the finite lifetime of nuclear states and inherent uncertainties in energy measurements, leading to a more realistic representation of the transition strength. By smoothing the spectral function, the Lorentzian improves the accuracy of calculated GT transition rates, allowing for a more direct comparison with experimental data and reducing discrepancies. Consequently, this method provides a more reliable interpretation of experimental results, strengthening the ability to probe fundamental nuclear properties and validate theoretical models used in nuclear physics research.

The study meticulously details a refinement of the generator coordinate method, specifically extending its capabilities to analyze Gamow-Teller transitions-a crucial aspect of understanding nuclear decay processes. This methodical approach mirrors a sentiment expressed by Mary Wollstonecraft: “It is time to try the method of reason.” The research doesn’t simply present calculations; it systematically builds a framework, projecting quantum numbers to improve the accuracy of nuclear matrix element calculations, and validating its results against established shell-model methods. This focus on reasoned methodology and validation exemplifies the core principle of building understanding through rigorous, step-by-step experimentation and analysis, revealing patterns in complex nuclear phenomena.

Where Do We Go From Here?

The extension of the projected generator coordinate method to Gamow-Teller transitions, as presented, offers a potentially valuable, albeit computationally demanding, path toward understanding nuclear matrix elements. The observed agreement with shell-model calculations is encouraging, yet serves as a reminder that validation against experimental data remains the ultimate test. It is worth noting that visual confirmation of convergence – the smooth evolution of results with increasing model space – requires patience; quick conclusions can mask structural errors.

A critical area for future investigation lies in systematically addressing the truncation schemes inherent to both the generator coordinate method and the configuration interaction shell model. The sensitivity of calculated matrix elements to these approximations needs careful scrutiny. Furthermore, a deeper exploration of the in-medium similarity renormalization group’s role in softening the divergences associated with infinite model spaces appears warranted. It is a question of balance – of choosing parameters that are mathematically tractable yet retain the essential physics.

Ultimately, the pursuit of accurate nuclear matrix elements for double-beta decay is not merely a technical exercise. It is an attempt to map the subtle patterns within the nucleus, to decipher the conditions under which stability falters, and to witness, perhaps, the rarest of transformations. The method described herein provides another lens through which to observe, and the resulting images demand careful interpretation.

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Original article: https://arxiv.org/pdf/2601.05058.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-10 01:34