Testing the Inverted Neutrino Mass Ordering with Neutrinoless Double-Beta Decay

We quantify the extent to which future experiments will test the existence of neutrinoless double-beta decay mediated by light neutrinos with inverted-ordered masses. While it remains difficult to compare measurements performed with different isotopes, we find that future searches will fully test the inverted ordering scenario, as a global, multi-isotope endeavor. They will also test other possible mechanisms driving the decay, including a large uncharted region of the allowed parameter space assuming that neutrino masses follow the normal ordering.

Neutrino oscillation [1][2][3][4] proves that the neutrino has mass and the Standard Model of particle physics is incomplete. The unknown origin of the neutrino mass has drawn enormous attention to neutrinoless doublebeta (0νββ) decay, a matter-creating nuclear transition in which two neutrons decay simultaneously into two protons, emitting only two new electrons and no antineutrinos [5]. The discovery of 0νββ decay would establish that the neutrino is its own antiparticle and has a Majorana mass [6]. It would also mark the first observation of a lepton-creating process, proving that neither lepton number (L) nor baryon minus lepton number (B − L) are symmetries of the Standard Model, as predicted by leading theories explaining the matter-antimatter asymmetry of our universe [7]. Indeed, searching for 0νββ decay is the most sensitive experimental approach to test Majorana neutrino masses and their associated L violation. It is also a unique probe of new physics at ultrahigh energy scales not accessible by current accelerators [8,9].
Different physics mechanisms can lead to 0νββ decay [10]. However, the exchange of light Majorana neutrinos interacting via standard, weak left-handed currents plays a special role. It is the only mechanism allowed by all theories predicting 0νββ decay, and typically dominates the rate of the process [11,12]. Assuming that the decay is mediated by light neutrinos, its half-life is [13] where G is the phase-space integral, g A 1.276 [14] where U ei are the elements of the full 6-parameter PMNS matrix [15].
Neutrino oscillation measurements constrain the range of allowed m ββ values [16] and prove that [17][18][19][20][21][22][23][24][25][26]. This implies that the effective Majorana mass is strictly larger than zero if neutrino masses follow the inverted ordering, i.e. m 3 < m 1 m 2 . In this case, the lowest m ββ value, minimized with respect to the unknown Majorana phases and m 3 , is given by with m 3 = U 2 e3 / U 2 e1 /m 1 − U 2 e2 /m 2 ≈ 3 meV. Using the latest values and uncertainties from the Particle Data Group [4], we obtain whose uncertainty is dominated by the uncertainty on the solar mixing angle θ 12 . Using the latest NuFIT results [27], we obtain (m min ββ ) IO = 18.6 ± 1.2 meV. The lower bound on m ββ corresponds to an upper bound on T 1/2 at the scale of 10 27 -10 28 years, depending on the value of the parameters in equation (1).
In the last decades, a vast experimental program has been mounted to develop experiments with sensitivity reaching (m min ββ ) IO , able to exhaustively test whether 0νββ decay is mediated by the exchange of light neutrinos with inverted-ordered masses [28,29]. Mature designs are now available for multiple detection techniques, and the physics community is discussing how to proceed. As part of this process, the Astroparticle Physics European Consortium (APPEC) is updating its 0νββ decay roadmap [29]   review. Conceptual designs are available for three experiments [30][31][32], whose construction can start as soon as funding is available. These experiments use different 0νββ-decaying isotopes and detection technologies, and can perform independent and complementary measurements.
As mentioned above, observing 0νββ decay would unambiguously demonstrate matter creation and prove the Majorana nature of neutrinos. However, the conversion between T 1/2 and m ββ in Eq. (1) is not trivial and requires inputs from nuclear theory. While the phase-space integral G has been calculated with negligible uncertainty [33], obtaining reliable nuclear matrix elements M is challenging, as it requires computationally intensive many-body calculations and the evaluation of several operators [13,34]. Four primary many-body methods have been historically used in the field: the nuclear shell model (NSM) [35][36][37], the quasiparticle random-phase approximation (QRPA) method [38][39][40][41][42], energy-density functional (EDF) theory [43][44][45], and the interacting boson model (IBM) [46,47]. For each of these methods, several calculations have been performed under different assumptions and approximations. The most recent results are listed in Table I. They can differ by up to a factor of three for a given isotope, and significant differences are present even within each method. The spread of values gives a rough idea of the many-body uncertainties on M (additional uncertainty contributions are discussed below). For some methods, calculations are not available for all isotopes.
The reach of 0νββ decay experiments is conventionally expressed in terms of discovery and exclusion sensitivities on m ββ . The discovery sensitivity corresponds to the smallest m ββ value for which an experiment has 50% probability of observing a signal at 99.7% confidence level (CL). The exclusion sensitivity corresponds to the median 90%-CL upper limit that an experiment will set on m ββ assuming 0νββ decay is not observable. As stated earlier, fully testing the inverted ordering scenario requires sensitivity to (m min ββ ) IO , accounting for its uncertainty. For discovery mode, this condition is met when the discovery sensitivity reaches the central value of (m min ββ ) IO : the (m min ββ ) IO uncertainty is symmetric, so the probability of lower or upper fluctuations is the same, and the 50% probability for an observation is preserved. However, for exclusion mode, the (m min ββ ) IO uncertainty reduces the CL by a variable amount that depends on the experimental parameters, mainly the background statistical uncertainty. Therefore the exclusion sensitivity on m ββ cannot be used to set an experiment-independent condition corresponding to fully covering the inverted ordering scenario. The discovery sensitivity is anyway the most appropriate metric for searches aiming to discover a process. Thus, reaching a discovery sensitivity of 18.4 meV is the right concrete goal for experiments aiming to explore the full inverted ordering parameter space. Figure 1 shows the discovery and exclusion sensitivities of proposed future experiments. We converted the T 1/2 sensitivity values quoted by the LEGEND [31], CU-PID [30] and nEXO [48] collaborations to m ββ values using the nuclear matrix elements of Table I. We group the calculations by many-body method to aid in the comparison of more consistent quantities. Remarkably, these experiments all show sensitivity to measure a 0νββ decay signal at the bottom of the inverted ordering parameter space. Some many-body methods, such as EDF theory and IBM, give systematically larger M values, pushing the m ββ sensitivity even lower. QRPA calculations on the other hand give a broad range of results, partially due to the role of nuclear deformation in this framework. The NSM provides M values which are typically smaller than for the other methods, but are not available for 100 Mo.
Comparing the performance of experiments using different isotopes is challenging because of the large uncertainties affecting the nuclear matrix element predictions. In particular, each many-body method uses different approximations, which are likely to result in a common over-or underestimation of the calculated M values. Even comparisons considering one many-body method at a time can raise concerns. The number of calculations available for each isotope and method can be significantly different, suggesting that not all many-body approaches are equally suitable for all isotopes. Consequently the range of M values cannot be quantitatively interpreted as the uncertainty, which currently remains unknown. One might be tempted to make comparisons based on the central value of the m ββ sensitivity within a specific nuclear method, but this value can be disproportionately affected by a single outlier matrix element. Weighted averages are also problematic, as the weight  [30,31,48], grouped by nuclear many-body frameworks with matrix element ranges from Table I given to each calculation would be to some extent arbitrary. Given the lack of objective criteria to compare experimental sensitivities in different isotopes, and the lack of a clear estimate of the uncertainties, we advocate to refrain from ranking experiments' reach quantitatively, and focus instead on the fact that we have a global, multiisotope endeavor that will fully test the inverted ordering scenario.
A broad effort to reduce uncertainties is ongoing within the nuclear theory community, with significant advances made in the last few years. Ab initio calculations that incorporate wider nuclear correlations and two-body currents have recently succeeded in predicting single β decay rates [49] with no need for "quenching" -an ad hoc reduction of the value of calculated matrix elements involving the nuclear spin required by less sophisticated calculations [13]. The first available ab initio 0νββ matrix element calculations in medium-sized nuclei, supported by studies in lighter systems [50,51], indicate a relatively mild suppression by tens of percent with respect to the lower limit of the range given in Table I [52]. Efforts are underway to improve the quality of these results, extend them to heavier nuclei, and include two-body currents at finite momentum transfers [53]. On the other hand, the contact term introduced in Refs. [34,54], which until recently went unrecognized, is a leading-order contribution to M . Effective field theory and ab initio nuclear structure provide a scheme for estimating this contribution [55,56]. A first study in 48 Ca suggests that this term can enhance M by about 40% percent [57], leading to a faster decay rate. In heavier systems, preliminary results suggest a roughly similar impact for all 0νββ isotopes, only slightly dependent on the nuclear many-body method [58]. Complementary studies using e.g. lattice QCD [59,60] will test whether this claimed enhancement is robust. If so, it may compensate the reduction in decay rate due to the inclusion of the "quenching" physics, leading to a picture similar to the one represented by Fig. 1. Thus, should the current theoretical results be confirmed, the proposed global 0νββ decay experimental effort would still fully probe the inverted ordering parameter space.
In this letter, we have focused on the inverted ordering scenario as a prominent goalpost for the proposed experimental 0νββ decay program. However, the discovery power of these experiments is high even assuming other, equally reasonable scenarios. By reaching a sensitivity of the order of tens of meV, these searches will probe a significant fraction of the remaining parameter space for left-handed neutrino exchange even if neutrino masses follow the normal ordering. Bayesian analyses suggest up to 50% discovery probabilities for the normal ordering scenario [61,62], and a non-vanishing discovery probability even assuming the most unfavorable value of the Majorana phases [63]. Significant advancement would also be made in probing the exchange of heavy mediators. For many such models, 0νββ decay searches probe energy scales beyond the reach of current accelerator technology [9]. Additional physics mechanisms could completely change the parameter space of interest, potentially even increasing the discovery power of future experiments [64][65][66][67]. In general, pushing 0νββ decay sensitivity to increasingly large half-life values explores uncharted territory, and new physics could manifest at any time.

ACKNOWLEDGMENTS
We would like to thank Christoph Wiesinger and Steven R. Elliott for valuable discussions. This work has been supported by the Science and Technology Fa-cilities Council (Grant No. ST/T004169/1), by the EU Horizon2020 research and innovation program under the Marie Sk lodowska-Curie Grant Agreement No. 754496, by the Spanish MICINN through the "Ramón y Cajal" program with grant RYC-2017-22781, the Italian Research Grant Number 2017W4HA7S "NAT-NET: Neutrino and Astroparticle Theory Network" under the program PRIN 2017 funded by MIUR, the AEI "Unit of Excellence María de Maeztu 2020-2023" award CEX2019-000918-M, the AEI grant FIS2017-87534-P and by the U.S. DOE Office of Nuclear Physics under Grant Number DE-FG02-97ER41020.