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The renormalization of the shell-model neutrinoless double-beta decay operator starting from effective field theory at next-to-next-to-leading order

This paper extends a previous study by constructing effective shell-model decay operators for neutrinoless double-beta decay in $^{48}$ Ca, $^{76}$ Ge, and $^{82}$ Se up to next-to-next-to-leading order, including for the first time three-body current contributions at next-to-next-to-next-to-leading order, to evaluate theoretical uncertainties arising from both model space limitations and chiral expansion truncation.

Original authors: S. L. Lyu, U. Carusone, G. De Gregorio, N. Itaco, L. Coraggio

Published 2026-10-26

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CC BY 4.0

Original authors: S. L. Lyu, U. Carusone, G. De Gregorio, N. Itaco, L. Coraggio

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 within the heart of matter, protons and neutrons bind together to form the atomic nucleus, a tiny, dense world governed by the strong force. For decades, physicists have tried to understand how these particles interact, often relying on models that fit the data but lack a deep connection to the fundamental laws of nature. A more rigorous approach has emerged, treating the nucleus not as a collection of mysterious blobs, but as a system built from the same rules that govern the subatomic particles called quarks and gluons. This framework, known as effective field theory, allows scientists to build a consistent picture of nuclear forces, starting from the most basic interactions and adding layers of complexity step by step. One of the most profound questions in this field involves a rare and hypothetical event called neutrinoless double-beta decay. In this process, two neutrons in a nucleus would simultaneously transform into two protons and emit two electrons, but without releasing the usual neutrinos. If observed, this decay would prove that neutrinos are their own antiparticles and could explain why the universe is made of matter rather than antimatter. However, detecting this event requires calculating the probability of it happening inside specific atoms with extreme precision, a task that has long been hindered by theoretical uncertainties.

A team of researchers has taken a significant step forward in this quest by refining the mathematical tools used to predict how likely this decay is to occur in three specific elements: calcium, germanium, and selenium. Their work focuses on the "operator," a complex mathematical description of the decay process that acts like a set of instructions for how the nucleus changes. In previous studies, scientists had only considered the most basic version of these instructions, which describe how two particles interact at a time. This new study expands that view by adding the next layers of complexity, accounting for interactions that involve three particles at once and including higher-order corrections that were previously ignored. The researchers built these instructions from the ground up using the same fundamental theory that describes the forces holding the nucleus together, ensuring that the description of the decay matches the description of the nucleus itself. By doing so, they created a fully consistent framework where the rules for the atom and the rules for its decay are derived from the same source, eliminating the guesswork that often plagues these calculations.

The team applied this new, more complete framework to calculate the nuclear matrix elements, which are essentially the numbers that determine how fast the decay would happen if it were possible. They focused on the ground states of the parent nuclei and their potential daughter nuclei, running detailed simulations to see how the results changed as they added more terms to their equations. The findings reveal a reassuring stability in the theory. When they added the more complex three-particle interactions, the predicted probability of the decay changed by less than one percent. This tiny shift suggests that the simpler, two-particle description is already capturing the vast majority of the physics at play. Furthermore, the researchers tested two different ways of handling the short-range, high-energy parts of the interaction, which are difficult to pin down. They found that regardless of which method they used, the final predictions for the decay probability remained very similar, differing by only a small amount that falls within the expected margin of error. This consistency strengthens the confidence that the current theoretical approach is sound and robust.

Beyond the specific numbers, the study provides a clear map of where the uncertainties lie. The researchers identified that the main source of potential error comes not from the fundamental laws of the decay, but from the limitations of the computer models used to simulate the nucleus. Even with the most advanced calculations, there is a small degree of approximation required to handle the thousands of particles involved. By carefully analyzing how their results changed as they improved the calculation step by step, they estimated that the total theoretical uncertainty for the most likely decay paths is around twenty percent. For the less common decay paths to excited states, the uncertainty is even smaller. This level of precision is a major achievement, as it tells experimentalists exactly how accurate their theoretical predictions are. It confirms that the leading terms in the theory are sufficient to describe the process and that the more complex, higher-order terms act as minor corrections rather than game-changers.

The implications of this work extend to the search for the decay itself. The researchers confirmed that for germanium and selenium, the decay to the ground state of the daughter nucleus is significantly more likely than decay to higher-energy states, making the ground-state transition the primary target for future experiments. For calcium, the probabilities for both transitions are similar, but the energy available for the decay to the higher state is lower, making it harder to observe. The study also highlights that the contribution of three-particle currents, which were once a major unknown, is negligible for these medium-sized nuclei. This finding simplifies the path forward for future research, allowing scientists to focus their efforts on refining the two-particle descriptions and reducing the uncertainties in the nuclear models themselves. By establishing a solid, consistent foundation for these calculations, the researchers have provided the community with a reliable benchmark. This clarity is essential for the next generation of experiments, which aim to detect this elusive decay and unlock the secrets of the neutrino's nature. The work does not solve the mystery of whether the decay happens, but it ensures that when the data arrives, scientists will have the most accurate theoretical tools possible to interpret it.

Technical Summary: Renormalization of the Shell-Model Neutrinoless Double- $\beta$ Decay Operator from Effective Field Theory

Problem Statement
The calculation of nuclear matrix elements ( $M^{0\nu}$ ) for neutrinoless double- $\beta$ ( $0\nu\beta\beta$ ) decay is critical for interpreting experimental limits on the effective neutrino mass. Historically, these calculations have relied on phenomenological nuclear Hamiltonians and transition operators. This paper addresses the need for a fully consistent framework where both the nuclear Hamiltonian and the decay operators are derived from the same underlying theory of the strong force: Chiral Effective Field Theory (ChEFT). Specifically, the authors aim to assess the theoretical uncertainties arising from two sources: the truncation of the chiral expansion of the decay operator (specifically at next-to-next-to-leading order, N2LO, and next-to-next-to-next-to-leading order, N3LO) and the truncation of the many-body perturbation theory (MBPT) used to derive effective shell-model (SM) operators. The study focuses on medium-mass candidates of experimental interest: $^{48}$ Ca, $^{76}$ Ge, and $^{82}$ Se.

Methodology
The authors employ a consistent ChEFT framework to construct the necessary inputs for shell-model calculations:

Nuclear Hamiltonian: The starting point is a nuclear Hamiltonian comprising two-nucleon (2NF) and three-nucleon (3NF) forces. The 2NF is derived from the Entem-Machleidt N3LO potential, while the 3NF is consistently built at N2LO. Both components share a non-local regulator with a cutoff $\Lambda = 500$ MeV.
Decay Operators: The $0\nu\beta\beta$ $0 ν β β$ decay operator is constructed via Chiral Perturbation Theory (ChPT) for the exchange of light Majorana neutrinos.
- Two-Body Currents: The operator includes contributions up to N2LO. This encompasses the leading order (LO) long-range component, the LO short-range contact term (regulated by the low-energy constant $g_{NN}^\nu$ ), and N2LO corrections (loop and ultrasoft contributions).
- Three-Body Currents: For the first time in this context, contributions from three-body currents emerging at N3LO are included.
Effective Operators: Many-body perturbation theory is used to derive effective shell-model Hamiltonians ( $H_{\text{eff}}$ ) and decay operators ( $\Theta_{\text{eff}}$ ) that account for configurations outside the truncated valence space. This involves the $\hat{Q}$ -box and $\hat{\Theta}$ -box formalisms. Crucially, the three-body components of the decay operator are treated by deriving a density-dependent two-body contribution via a normal-ordering procedure to make them compatible with standard SM codes.
Renormalization of $g_{NN}^\nu$ : The study evaluates two approaches to fix the LEC $g_{NN}^\nu$ $g_{N N}^{ν}$ associated with the short-range contact term:
- Relating it to charge-independence-breaking (CIB) contact interactions (local regulator).
- Inferring it from $nn \to ppe^-e^-$ transition amplitude calculations using the Entem-Machleidt N3LO interaction (non-local regulator).
Uncertainty Quantification: Theoretical uncertainties are estimated using:
- MBPT Truncation: Comparing third-order MBPT results with Padé approximants $[2|1]$ .
- Chiral Truncation: Estimating residual errors based on the order-by-order convergence of the ChPT expansion (using a breakdown scale $\Lambda = 500$ MeV and typical momentum $p \approx 200$ MeV).

Key Contributions

First Inclusion of N3LO Three-Body Currents: This work introduces the contribution of three-body currents (derived at N3LO in ChPT) into the calculation of $M^{0\nu}$ for nuclei of experimental interest ( $^{48}$ Ca, $^{76}$ Ge, $^{82}$ Se).
Consistent Framework: Unlike previous studies that used phenomenological Hamiltonians with ChPT-derived operators, this work maintains consistency between the Hamiltonian and the decay operator, both derived from the same ChPT expansion.
Systematic Uncertainty Analysis: The paper provides a rigorous decomposition of theoretical uncertainties, separating those arising from the many-body expansion from those arising from the chiral expansion of the operator.
Regulator Independence Check: The study demonstrates that the choice between local and non-local regulators for the short-range contact term yields consistent $M^{0\nu}$ values within theoretical uncertainties.

Results

Perturbative Convergence: The perturbative expansion of the effective SM operator shows good convergence. The Gamow-Teller component dominates the matrix elements, while Fermi and Tensor components are less sensitive to higher orders.
Impact of Higher Orders:
- Moving from Leading Order (LO) to N2LO for two-body currents results in changes of approximately 7–11% for ground-state transitions.
- The inclusion of N3LO three-body currents contributes less than 1% to the total $M^{0\nu}$ , confirming that two-body currents are the dominant contribution in a $\Delta$ -less approach.
Calculated Matrix Elements: Using a non-local regulator and including 3BC, the calculated Padé approximant values for ground-state transitions are:
- $^{48}$ Ca $\to$ $^{48}$ Ti: $0.82$
- $^{76}$ Ge $\to$ $^{76}$ Se: $1.64$
- $^{82}$ Se $\to$ $^{82}$ Kr: $1.31$
Uncertainty Estimates:
- Uncertainties from MBPT truncation (Padé vs. 3rd order) are estimated at roughly 10–15% for ground-state transitions.
- Uncertainties from chiral truncation (LO to N2LO) are smaller, not exceeding 7%.
- The total combined theoretical uncertainty is approximately 20% for ground-state transitions.
Yrare States: Matrix elements for transitions to yrare $0^+$ states are generally an order of magnitude smaller than those to the ground state (except for $^{48}$ Ca where they are comparable but suppressed by Q-value differences), making these decays unlikely to be observed.

Significance and Claims
The authors claim that their results validate the robustness of the ChEFT approach for defining the $0\nu\beta\beta$ decay operator. The small impact of N3LO three-body currents and the consistency between different regulator choices suggest that the LO and N2LO two-body currents provide a sound definition of the operator for medium-mass nuclei. The study establishes that the dominant theoretical uncertainties are currently driven by the many-body perturbative expansion rather than the truncation of the chiral expansion of the operator itself. The calculated values for $^{76}$ Ge are consistent with recent ab initio emulator results within their respective theoretical error bars, supporting the reliability of the shell-model approach when grounded in ChEFT. The authors position this work as a step toward future analyses of heavier systems ( $^{100}$ Mo, $^{130}$ Te, $^{136}$ Xe) and benchmarking against ab initio methods.

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