Neutrinoless Double Beta Decay

• Neutrinoless double beta decay is a rare nuclear process that violates lepton number and, if observed, would confirm neutrinos as their own antiparticles.
• Accurate computation of nuclear matrix elements and phase-space factors links experimental half-life limits to the effective Majorana neutrino mass.
• Cutting-edge experimental techniques and theoretical advances aim to probe sensitivities down to 10–20 meV, offering insights into new physics.

Neutrinoless double beta decay (0νββ) is a lepton-number-violating nuclear transition, $(A,Z)\to(A,Z+2)+2e^-$, in which two neutrons within a nucleus convert to two protons with the emission of only two electrons and no (anti)neutrinos. This process is forbidden in the Standard Model, where lepton number is conserved and only two-neutrino double beta decay is allowed. Observing 0νββ would establish that neutrinos are Majorana fermions (i.e., particles identical to their own antiparticles), demonstrate total lepton number violation, and provide information about the absolute scale of neutrino masses and the mechanism by which they acquire mass (Adams et al., 2022, Jones, 2021, Kim, 2020, Dolinski et al., 2019, Cardani, 2018).

1. Theoretical Foundations

Neutrinoless double beta decay violates lepton number by two units (ΔL = 2). The underlying physical principle links the observation of this decay to the Majorana character of the neutrino, as established by the Schechter–Valle theorem: if any mechanism permits 0νββ, there must exist at least a nonzero Majorana mass term for the light neutrinos (Adams et al., 2022, Deppisch et al., 2012). In the simplest "mass mechanism," the decay proceeds via the exchange of a virtual light Majorana neutrino between two vertices mediated by the Standard Model $V$–$A$ weak current. The neutrino propagator couples the two $W$ vertices through the effective Majorana neutrino mass: $$m_{\beta\beta}\equiv\bigg|\sum_{i=1}^3 U_{ei}^2\, m_i\bigg|$$ where $m_i$ are neutrino mass eigenstates and $U_{ei}$ are elements of the PMNS matrix, including complex Majorana phases (Kim, 2020, Jones, 2021, Adams et al., 2022).

Once integrated over the kinematics, the inverse half-life for an isotope undergoing 0νββ is

$$\left(T_{1/2}^{0\nu}\right)^{-1} = G^{0\nu} |M^{0\nu}|^2\left(\frac{m_{\beta\beta}}{m_e}\right)^2$$

Here $G^{0\nu}$ is a phase-space factor, $M^{0\nu}$ is the nuclear matrix element (NME), and $m_e$ is the electron mass (Grebe, 1 Apr 2025, Dolinski et al., 2019, Vergados et al., 2012). This rate directly links 0νββ to the absolute scale and character of neutrino mass.

2. Nuclear Matrix Elements and Phase-Space Factors

The nuclear matrix element, $M^{0\nu}$, encodes all nuclear structure uncertainties and carries the largest theoretical ambiguity in 0νββ predictions. It is comprised of Fermi (F), Gamow–Teller (GT), and tensor (T) components: $$M^{0\nu} = M_{GT}^{0\nu} - \left(\frac{g_V}{g_A}\right)^2 M_F^{0\nu} + M_T^{0\nu}$$ where $g_A$ and $g_V$ denote the axial and vector couplings. The evaluation involves summing two-body transition operators over all nucleon pairs; the neutrino potential depends on closure energy and internucleon separation (Grebe, 1 Apr 2025, Faessler, 2011, Vergados et al., 2012).

Methods for $M^{0\nu}$ computation include:

• Interacting Shell Model (ISM): full correlations in a truncated valence space; yields smaller $M^{0\nu}\sim2$–$3$.
• Quasiparticle Random Phase Approximation (QRPA): large single-particle space, collective correlations; typically $M^{0\nu}\sim4$–$6$.
• Interacting Boson Model (IBM-2): collective S, D bosons; $M^{0\nu}\sim4$–$5$.
• Energy Density Functional (EDF)/Generator Coordinate Method (GCM): mixes mean-field solutions; $M^{0\nu}\sim4$–$6$.

Ab initio approaches based on nuclear effective field theory (EFT) and lattice QCD are being developed to constrain short-range contributions and reduce $M^{0\nu}$ uncertainty below the current $\sim$30–50% (Grebe, 1 Apr 2025, Adams et al., 2022). The phase-space factor $G^{0\nu}$, which depends on the nuclear $Q$-value and atomic number, is precisely calculated and typically $\sim 10^{-14}$–$10^{-15}$ yr$^{-1}$ for leading isotopes (Adams et al., 2022, Brugnera, 17 Jan 2025).

3. Experimental Searches and Techniques

State-of-the-art 0νββ searches deploy diverse isotopes and technologies to maximize sensitivity and control backgrounds:

Isotope	Q$_{\beta\beta}$ (MeV)	Key Experiments	Typical NME Range
$^{76}$Ge	2.039	GERDA, Majorana, LEGEND	2.8–6.1
$^{136}$Xe	2.458	KamLAND-Zen, EXO-200, nEXO, NEXT	1.6–4.0
$^{130}$Te	2.527	CUORE, SNO+, CUPID	2.5–5.0

• High-purity Ge detectors (e.g., LEGEND): outstanding energy resolution ($\sim0.1$% FWHM at $Q$), background indices $<5$ counts/(ton·yr) (Brugnera, 17 Jan 2025).
• Liquid/gaseous xenon TPCs (EXO-200, nEXO, NEXT): large masses, topological event discrimination, self-shielding, and potential for Ba$^{++}$ tagging.
• Bolometric TeO$_2$/Mo/Se crystals (CUORE, CUPID): low temperatures, particle ID, strong rejection of $\alpha$ backgrounds.
• Large-scale liquid scintillator with dissolved Xe/Te (KamLAND-Zen, SNO+): high mass, moderate energy resolution.

The most stringent half-life limits (90% CL) are:

• $$T_{1/2}^{0\nu}(^{136}\mathrm{Xe}) > 1.07 \times 10^{26}$$ yr (KamLAND-Zen)
• $$T_{1/2}^{0\nu}(^{76}\mathrm{Ge}) > 1.8\times10^{26}$$ yr (GERDA + Majorana)
• $$T_{1/2}^{0\nu}(^{130}\mathrm{Te}) > 1.5\times10^{25}$$ yr (CUORE)

These correspond, given $|M^{0\nu}|$ uncertainties, to $m_{\beta\beta}$ upper bounds in the $60$–$200$ meV range (Jones, 2021, Brugnera, 17 Jan 2025, Adams et al., 2022).

4. Particle Physics Implications and New Physics Mechanisms

While the mass mechanism dominates in minimal scenarios, 0νββ provides a generic test of any $\Delta L=2$ process. The black-box (Schechter–Valle) theorem guarantees a nonzero Majorana mass whenever 0νββ is observed—even if new short-range mechanisms dominate (Deppisch et al., 2012, Adams et al., 2022). Other possible amplitudes include:

• Heavy right-handed neutrino exchange (left–right symmetric models): gives contact operators, amplitude scales as $1/M_N$.
• R-parity-violating supersymmetric interactions: gluino or neutralino exchange yields $\sim 1/M_{\mathrm{SUSY}}^5$ suppressed contributions (Deppisch et al., 2012).
• Scalar or vector leptoquarks, extra-dimensional sterile neutrinos, and Majoron emission (accompanying scalar emission).

Combinations of half-lives from multiple isotopes, and event kinematics (individual electron energies, angular distributions), allow discrimination among mechanisms (Deppisch et al., 2012, Graf et al., 2020). For example, right-handed current or scalar operators yield distinct electron angular correlations as compared to the mass mechanism.

5. Uncertainties, Nuclear Theory Developments, and the Role of Lattice QCD

The translation between an observed or constrained $T_{1/2}^{0\nu}$ and $m_{\beta\beta}$ is dominated by the NME uncertainty, which translates to $\sim$50% spread in $m_{\beta\beta}$ interpretation (Grebe, 1 Apr 2025, Faessler, 2011, Dolinski et al., 2019). This "theory bottleneck" motivates:

• Advanced QRPA, shell-model, IBM, and GCM methods, with benchmarking to data from 2νββ decays and single-$\beta$ transitions.
• Inclusion of ab initio nuclear many-body approaches, leveraging chiral EFT and lattice QCD–derived low-energy constants (LECs) for short-range operator matching.
• Systematic quantification of uncertainties: configuration space, short-range correlations, $g_A$ quenching, nuclear deformation, and operator renormalization.

Direct lattice QCD calculations of 0νββ in heavy nuclei remain out of reach, but calculations of LECs for chiral EFT and pion–nucleon couplings are under active development and have already constrained relevant two-nucleon contact terms and pion-exchange contributions at the $\lesssim$20% level (Grebe, 1 Apr 2025).

6. Future Prospects, Discovery Reach, and the Neutrino Mass Hierarchy

Next-generation experiments target $T_{1/2}^{0\nu}\gtrsim 10^{27-28}$ yr (e.g., LEGEND-1000, nEXO, CUPID), probing $m_{\beta\beta}$ down to $10$–$20$ meV and thus completely covering the inverted mass ordering parameter space (Brugnera, 17 Jan 2025, Adams et al., 2022, Garfagnini, 2014). In the inverted ordering, $m_{\beta\beta}$ cannot be cancelled by Majorana phase alignment and so a significant lower bound (typically $15$–$50$ meV) exists, while for normal hierarchy the rate may be unobservably suppressed by destructive interference among the three neutrino mass eigenstates' contributions (Bilenky et al., 2012, Rodejohann, 2010).

A robust positive signal in multiple isotopes would allow for disentanglement of operator mechanism and precise extraction of $m_{\beta\beta}$, while improving bounds would inform on the possible neutrino mass spectrum and on the viability of baryogenesis via leptogenesis (Adams et al., 2022, Dolinski et al., 2019).

7. Conclusion

Neutrinoless double beta decay serves as the principal probe for lepton number violation and the Majorana character of neutrinos at accessible energies. Observation of 0νββ would provide not only the first evidence of physics beyond the Standard Model but also direct information about the origin and absolute scale of neutrino mass. The field is characterized by rapid experimental advances toward the ton-scale and complementary theoretical efforts to control nuclear matrix element uncertainties. Achieving an unambiguous mapping between any future positive signal and fundamental neutrino properties will require continued progress across both experimental and nuclear theory fronts, emphasizing multi-isotope, multi-technique confirmation and systematic uncertainty reduction (Adams et al., 2022, Brugnera, 17 Jan 2025, Grebe, 1 Apr 2025, Jones, 2021, Dolinski et al., 2019).