Lepton Number Violation Overview

• Lepton Number Violation is a phenomenon where the total lepton count changes (typically by 2), indicating the possible Majorana nature of neutrinos and providing insights into neutrino mass generation.
• Theoretical frameworks employ the Weinberg operator and higher-dimensional SMEFT operators, which connect neutrino mass mechanisms to observable phenomena in rare nuclear and meson decays.
• Experimental probes range from neutrinoless double beta decay to same-sign dilepton signatures at colliders, offering complementary avenues to test and distinguish LNV models.

Lepton number violation (LNV) encompasses processes in which the total lepton number, typically an accidental global symmetry in the Standard Model (SM), changes in particle interactions. While lepton number conservation is experimentally robust at the renormalizable level within the SM, numerous ultraviolet completions and extensions predict its violation. Observation of LNV would have far-reaching implications: it would signal the Majorana nature of neutrinos, illuminate the origin of neutrino masses, and impact understanding of baryogenesis mechanisms. LNV may manifest at low energies in rare nuclear processes such as neutrinoless double beta decay, in flavor-changing meson decays, or at high energies through distinctive collider signatures involving same-sign leptons. This article systematically reviews theoretical mechanisms for LNV, its manifestation in neutrino mass generation, the role of effective field theory operators, phenomenological signatures across experimental frontiers, and the broader implications for particle physics.

1. Fundamental Mechanisms for Lepton Number Violation

LNV is realized when interactions break the global $U(1)_L$ symmetry by $\Delta L \neq 0$, most commonly by two units ($\Delta L = 2$), though higher multiples such as $\Delta L = 3$ are also considered. The most generic source of $\Delta L = 2$ is the dimension-five Weinberg operator,

$O_5 = (L H) (L H) / \Lambda,$

where $L$ is the lepton doublet, $H$ is the Higgs doublet, and $\Lambda$ is the scale of new physics. This operator, upon electroweak symmetry breaking, leads to Majorana neutrino masses. Subsequent ultraviolet completions generate $O_5$ via tree-level (for example, type-I, II, III seesaw models) or loop-level (radiative) mechanisms.

Beyond the minimal Weinberg operator, higher-dimensional operators in the Standard Model Effective Field Theory (SMEFT), notably at dimension seven and nine, give rise to LNV processes not directly linked to neutrino masses—such as rare meson decays or $\mu \to e^+$ conversion—which may violate both lepton number and lepton flavor (Fridell et al., 2023, Berryman et al., 2016).

Exotic mechanisms where lepton number is violated by higher units, such as $\Delta L = 3$, have also been constructed, with the lowest-order effective operators arising at dimension nine or higher (Fonseca et al., 2018, Fonseca, 2019). Gauge symmetry structure and discrete symmetries (e.g., $Z_3(L)$, $U(1)_{3B-L}$) are crucial in determining the allowed pattern of LNV.

2. LNV and Neutrino Mass Generation

LNV is intricately related to the origin of neutrino masses. Majorana neutrinos, arising from LNV, result in mass terms of the form

$$\mathcal{L}_\nu \supset \frac{1}{2} m_\nu \nu^T C \nu + \mathrm{h.c.}$$

This mass term violates lepton number by two units. The canonical type-I seesaw incorporates heavy right-handed neutrinos $N$ with Majorana masses, leading to small masses for the active neutrinos by seesaw suppression. Variants such as the inverse seesaw or conformal inverse seesaw (Humbert et al., 2015) introduce additional singlet fermions and small lepton number breaking parameters, enabling observable LNV even when active neutrino masses are suppressed.

Radiative mechanisms, such as those mediated by colored scalars or leptoquarks (Perez et al., 2010, Kohda et al., 2012, Cata et al., 2019), generate LNV neutrino masses via loop diagrams involving new colored states. In 3-3-1 models, spontaneous or explicit LNV yields Majorana masses, but in some constructions (e.g., the economical model) the would-be Majoron is gauged away, embedding LNV in the gauge sector (Benavides et al., 2015).

A characteristic of linear seesaw models is that the mass splitting of heavy quasi-Dirac neutrinos, responsible for LNV, is directly proportional to the light neutrino mass differences, with oscillation lengths of heavy neutrino–antineutrino coherence set by $\Delta M \sim \Delta m^2_\nu$ (Batra et al., 2023).

3. Effective Operators and Flavor Structure

The SMEFT provides a systematic categorization of LNV operators by dimension. At dimension five, only the Weinberg operator arises. At dimension seven, twelve independent $\Delta L = 2$ operators exist, including classes with additional Higgs fields, covariant derivatives, or field-strength tensors (Fridell et al., 2023). At dimension nine, four-fermion operators encoding $\Delta L = 2$ appear, such as those relevant to neutrinoless double beta decay (0$\nu\beta\beta$).

The flavor structure of these operators is of crucial importance. For instance, operators antisymmetric in flavor indices may have vanishing $ee$ (electron) entries, suppressing 0$\nu\beta\beta$, but contributing to off-diagonal transitions such as $\mu^- \to e^+$ (Berryman et al., 2016). This underscores the complementarity between different experimental searches targeting various lepton flavor final states.

At even higher operator dimension, LNV by three units ($\Delta L = 3$) is mediated by operators such as

$$O_9^{(1)} = u^c u^c u^c e^c L L, \qquad O_{13} = u^c u^c u^c u^c d^c e^c e^c e^c,$$

leading to rare proton decay channels involving three leptons (Fonseca et al., 2018, Fonseca, 2019). The high dimensionality implies a strong suppression by powers of the new physics scale $\Lambda$ (e.g., $\Gamma_p \sim m_p^5/\Lambda^{18}$ for a dimension-13 operator), making such LNV observable only if $\Lambda$ is near the TeV scale.

4. Experimental Probes across Frontiers

4.1. Nuclear and Meson Experiments

The gold standard for LNV searches is 0$\nu\beta\beta$ decay, with half-lives constrained to be $T_{1/2} > 10^{26}$ yr for certain isotopes. This process directly probes the effective Majorana mass $m_{\beta\beta}$, related to the sum of light neutrino mass eigenstates weighted by mixing matrix elements and CP phases. However, non-observation may be due to specific texture zeros or flavor structure in the LNV operator (Berryman et al., 2016, Fridell et al., 2023).

Rare decays of mesons ($K \to \pi \nu\nu$, $B \to K^{(*)}\nu\nu$, $D \to K \ell \ell \pi$) with LNV final states provide complementary constraints, especially on operators involving flavor off-diagonal couplings (Delepine et al., 2011, Dong et al., 2013).

4.2. Collider Signatures

Collider experiments probe LNV via signatures not accessible at low energies. At hadron colliders (LHC), relevant channels involve

• Same-sign dileptons plus jets: $pp \to \ell^\pm\ell^\pm + X$, arising from heavy Majorana neutrino exchange or decays of colored octets or leptoquarks (Perez et al., 2010, Kohda et al., 2012, Benavides et al., 2015, Li et al., 2021).
• Multi-lepton final states: In $\Delta L=3$ models, three same-sign leptons plus jets, or five-body proton decays, become viable probes (Fonseca et al., 2018, Fonseca, 2019).
• Long-lived particle searches: Scenarios with suppressed couplings can render intermediate states long-lived, enabling detection via displaced vertices at ATLAS/CMS or far detectors like MATHUSLA (Li et al., 2021).

Muon and electron colliders open further opportunities. Same-sign lepton colliders (e.g., $\mu^+\mu^+$ at $\mu$TRISTAN) provide background-free environments for testing LNV, with sensitivity to both heavy neutral leptons (HNLs) and neutrinophilic scalars (Lima et al., 22 Nov 2024). At future muon colliders, vector boson fusion can produce Higgs bosons with subsequent LNV decays ($h/H \to NN$) to observable same-sign lepton plus jets final states (Yang et al., 12 May 2025).

4.3. Summary Table: LNV Signatures

Experimental Context	Key LNV Signature	Relevant Operator/Mechanism
0νββ decay (nuclear)	$ee$ final state (ΔL=2)	Weinberg operator, $O_5$
Meson/top decays	$\ell^+\ell^+/l^-\ell^-$	heavy Majorana ν exchange (resonant)
LHC/same-sign leptons	$\ell^\pm\ell^\pm$ + jets	colored octet, leptoquark, HNL, SMEFT d=7/9
Muon colliders, VBF Higgs	$\mu^\pm\mu^\pm$ + fat-jets	Higgs-singlet mixing + seesaw (Yang et al., 12 May 2025)
Proton decay (ΔL=3)	$3\ell^+$ + mesons	d=13 operators (Fonseca et al., 2018, Fonseca, 2019)

5. Phenomenological Impacts and Model Discrimination

Several critical aspects determine the phenomenology of LNV:

• Parameter Correlation: For many scenarios, such as the minimal linear seesaw, the observable LNV rates (e.g., ratio $R_{\ell\ell}$ of same-sign to opposite-sign leptons) are directly related to neutrino mass parameters—mass ordering, mass splitting, and CP phases (Batra et al., 2023, Gluza et al., 2016).
• Flavor Structure: The distribution of LNV across flavors (e.g., $ee$, $e\mu$, $\mu\mu$) is pivotal for extracting underlying operator coefficients and for distinguishing between SMEFT operators (Fridell et al., 2023, Berryman et al., 2016).
• Suppression Mechanisms: In models with a large number of SM copies, unitarity and permutation symmetry can enforce cancellations in LNV amplitudes ($\sum_k (U_{1k})^2 m_k = 0$) resulting in negligible rates (Kovalenko et al., 2010).
• Resonant Enhancement: When the intermediate state (e.g., heavy Majorana neutrino) is kinematically accessible and can go on-shell, rates for LNV processes such as meson or top decays can be significantly enhanced (Delepine et al., 2011, Dong et al., 2013).
• Operator Scaling: Collider rates for dimension-7 SMEFT operators scale as $\Lambda^{-6}$, giving colliders complementary reach to nuclear/low-energy experiments (Fridell et al., 2023).

Combining results from low-energy, flavor, and collider observables allows for the disambiguation of models and identification of dominant operator contributions, particularly when signals are observed in more than one experimental context.

6. Outlook and Theoretical Implications

The observation of lepton number violation would establish the Majorana character of neutrinos and confirm physics beyond the Standard Model. It would discriminate among neutrino mass mechanisms, constrain or favor classes of ultraviolet completions, and map the flavor structure of new interactions. Experiments in the current and next generation—across low-energy nuclear physics, high-intensity meson factories, hadron and lepton colliders—are sensitive to overlapping and distinct parts of LNV parameter space.

Collaborative analyses exploiting the synergy of different observables (e.g., $0\nu\beta\beta$ decay, $\mu^-\to e^+$ conversion, LHC same-sign dileptons, rare multi-lepton proton decays, precision Higgs studies at muon colliders) are crucial for a comprehensive characterization of LNV and for isolating its theoretical origin. The detection or stringent limits on LNV processes will continue to shape the development of models for neutrino mass, baryogenesis via leptogenesis, and the broader landscape of physics beyond the Standard Model.