Solution to the uncertainty problem of nuclear matrix element for neutrinoless double-$\beta$ decay

Technical Summary: A Solution to the Uncertainty Problem of the Nuclear Matrix Element for Neutrinoless Double-β Decay

Problem Statement
The detection of neutrinoless double-β decay ($0\nu\beta\beta$) is a primary objective in neutrino physics, as it would confirm the Majorana nature of neutrinos and extend the Standard Model. However, the extraction of the effective neutrino mass ($\langle m_\nu \rangle$) from experimental half-life limits relies critically on the accuracy of the Nuclear Matrix Elements (NMEs). For over 40 years, theoretical calculations of NMEs have exhibited large uncertainties, with results from different nuclear structure models (e.g., QRPA, Shell Model, IBA) distributed over a broad range. This dispersion has hindered the progress of neutrino physics, as the uncertainty in NMEs translates directly into uncertainty in the derived neutrino mass.

Methodology
The authors propose a method to reduce this uncertainty by leveraging the relationship between the $0\nu\beta\beta$ decay and the two-neutrino double-β decay ($2\nu\beta\beta$). The core of their approach involves:

1. Perturbed Transition Operator: The authors utilize a calculation framework where the NMEs are derived using the perturbed transition operator. This includes higher-order terms beyond the leading-order (first-order Rayleigh-Schrödinger perturbation), specifically vertex corrections and two-body current terms (referred to as RS two-body current terms).
2. Effective Axial-Vector Coupling ($g_A^{eff}$): They define an effective axial-vector coupling constant, $g_A^{eff}$, which encapsulates many-body effects not fully included in leading-order NMEs. This parameter is determined by equating the half-life calculated with leading-order NMEs and $g_A^{eff}$ to the half-life calculated with perturbed NMEs and the bare coupling $g_A^{bare}$.
3. Similarity Relation: A key finding from their previous work is that the theoretical $g_A^{eff}$ values for the $0\nu\beta\beta$ and $2\nu\beta\beta$ modes are remarkably similar, despite the vast difference in neutrino momentum between the two processes ($\sim 100$ MeV/c vs. $\sim 2-3$ MeV/c). This similarity arises because the neutrino potential, present in both the numerator and denominator of the ratio defining $g_A^{eff}$ for the $0\nu\beta\beta$ mode, approximately cancels out.
4. Renormalization Procedure: Since the $2\nu\beta\beta$ half-life is experimentally measured, the authors determine $g_A^{eff}$ for the $2\nu\beta\beta$ mode ($g_{A,2\nu}^{eff}$) that reproduces this experimental value. They then apply this value to the $0\nu\beta\beta$ calculation using a renormalization factor derived from the ratio of theoretical $g_A^{eff}$ values. This allows the inclusion of non-perturbative many-body effects in the $0\nu\beta\beta$ NME without requiring the full convergence of higher-order perturbative terms.

Application and Results
The method was applied to four isotopes: $^{136}\text{Xe}$, $^{76}\text{Ge}$, $^{130}\text{Te}$, and $^{110}\text{Pd}$. The authors collected NMEs from various groups using different nuclear structure methods (QRPA, Shell Model, IBA) and applied their renormalization procedure.

• Reduction of Dispersion: The most significant result is the dramatic reduction in the dispersion of the NMEs. For $^{136}\text{Xe}$, the range of the modified NMEs (multiplied by $g_A^2$) was reduced to one-fourth of the original interval. For $^{76}\text{Ge}$, the dispersion of the deformed QRPA results was reduced by 89%, and the overall dispersion for all methods was reduced by approximately 40–50%.
• Lowering of NME Values: In many cases, the modified NME values were lowered compared to previous calculations. For instance, the mean value for $^{76}\text{Ge}$ was lowered by 2.7 units.
• Effective Neutrino Mass ($\langle m_\nu \rangle$): Using the modified NMEs and current experimental lower limits for the half-life, the authors calculated the implied $\langle m_\nu \rangle$.
  • For $^{136}\text{Xe}$, the results indicate that the extracted $\langle m_\nu \rangle$ does not yet reach the inverted-mass-hierarchy region (assuming the lightest neutrino mass is $< 10$ meV).
  • For $^{76}\text{Ge}$ and $^{130}\text{Te}$, similar trends were observed. The modified calculations suggest that current experimental limits may not be sufficient to probe the inverted hierarchy region, unlike what some previous, higher NME estimates might have suggested.
• Systematics: The study confirmed that the ratio of $g_{A,0\nu}^{eff}$ to $g_{A,2\nu}^{eff}$ is close to unity (converging at first order) across different isotopes, validating the method's applicability. The analysis of correction terms (vertex and two-body currents) showed strong mass-number ($A$) dependence, particularly for the two-body current term, which explains the varying degrees of cancellation observed in different nuclei (e.g., strong cancellation in $^{76}\text{Ge}$ vs. large corrections in $^{136}\text{Xe}$).

Significance and Claims
The paper claims to offer a "possible solution" to the longstanding uncertainty problem of $0\nu\beta\beta$ NMEs. The significance of this work lies in:

1. Phenomenological Calibration: It provides a method to incorporate many-body effects non-perturbatively by calibrating against the experimentally known $2\nu\beta\beta$ decay, rather than relying solely on theoretical convergence of higher-order terms.
2. Universality: The method is applicable to results obtained by different groups using different nuclear structure models, provided the leading-order NMEs are calculated consistently.
3. Reduced Uncertainty: By significantly narrowing the dispersion of NME values, the method reduces the theoretical uncertainty in the extraction of the effective neutrino mass from experimental data.
4. Realistic Prospects: The results suggest a more conservative outlook on the observability of the inverted mass hierarchy region with current and near-future experiments, as the modified NMEs yield lower values for the decay rate compared to some previous estimates.

The authors conclude that their approach represents a "unique new direction" toward resolving the NME uncertainty, supported by the robustness of the similarity relation between the effective couplings of the two decay modes.