Lepton number

In particle physics, the lepton number is a conserved additive quantum number within the Standard Model that distinguishes leptons from other fundamental particles, with leptons (electrons, muons, taus, and their neutrinos) assigned +1 and antileptons -1, while all other particles have 0; it was introduced in 1953 to explain the absence of certain observed decay modes in weak interactions.^[1] The total lepton number L is defined as the sum of the individual lepton family numbers: L = L_e + L_\mu + L_\tau, where each family (electron, muon, and tau) conserves its own lepton number separately in the Standard Model assuming massless neutrinos.^[1] This conservation law ensures that processes like beta decay maintain balance, such as in n \to p + e^- + \bar{\nu}_e, where the initial lepton number of 0 equals the final (antineutrino contributes -1, electron +1).^[1] While strictly conserved at tree level in the Standard Model, lepton number can be violated by higher-dimensional effective operators, notably the dimension-5 Weinberg operator (\Delta L = 2), which generates Majorana masses for neutrinos and is implicated in neutrinoless double-beta decay ($0\nu\beta\beta), with current experimental limits as of 2025 placing the half-life beyond $10^{26} years (exceeding $2 \times 10^{26} years for isotopes like ^{76}Ge) for certain isotopes.^[1]^[2] Lepton flavor violation, observed in neutrino oscillations, mixes the family numbers but preserves total L to high precision, with searches for charged-lepton flavor violation (e.g., \mu \to e\gamma) yielding branching ratios below $1.5 \times 10^{-13} (90% CL) as of 2025.^[1]^[3] Beyond the Standard Model, lepton number violation is a key probe for new physics, such as seesaw mechanisms for neutrino masses or grand unified theories, where it may connect to baryon number violation and matter-antimatter asymmetry in the universe.^[4] Experimental efforts at colliders like the LHC and underground detectors continue to test these limits, potentially revealing extensions to the Standard Model if violations are detected.^[1]

Basic Concepts

Definition

In particle physics, the lepton number L is defined as an additive quantum number that quantifies the difference between the number of leptons and antileptons in a given process or system: L = n_l - n_{\bar{l}}, where n_l counts the leptons and n_{\bar{l}} counts the antileptons.^[1] This quantum number serves to distinguish leptons from other fundamental particles, such as quarks, by assigning leptons a value of +1 and antileptons a value of -1, while non-leptonic particles receive 0.^[1] Leptons are a class of elementary fermions that do not experience the strong nuclear force, encompassing the charged leptons—the electron (e), muon (\mu), and tau (\tau)—along with their neutral counterparts, the neutrinos (\nu_e, \nu_\mu, \nu_\tau). Within the Standard Model of particle physics, the total lepton number is conserved across all interactions, including the weak interactions that govern processes involving leptons, due to an underlying global U(1) symmetry.^[1] This conservation manifests in reactions as \Delta L = 0, ensuring the net lepton number remains unchanged before and after the interaction.^[1]

Historical Development

The concept of lepton number was introduced in 1953 by Emil J. Konopinski and Hormoz Mahmoud in their formulation of the universal Fermi interaction describing beta decay processes. They assigned a conserved quantum number, termed the "lepton charge," with a value of +1 to electrons and electron neutrinos, and -1 to their antiparticles, ensuring that weak interactions preserved this quantity. This postulate provided a systematic way to account for the observed conservation patterns in beta decays, such as neutron decay (n → p + e⁻ + \bar{ν}_e), where the total lepton number remains zero, while forbidding processes that would violate it, like the unobserved decay of a neutron directly into a proton and electron without a neutrino. This development occurred amid efforts to experimentally verify the neutrino's existence, postulated by Wolfgang Pauli in 1930 to resolve the continuous energy spectrum in beta decay. The Cowan–Reines experiment, initiated in 1953 at the Hanford reactor and refined in 1956 at Savannah River, detected antineutrinos through inverse beta decay (\bar{ν}_e + p → n + e⁺), providing direct evidence for the neutrino and aligning with the lepton number conservation rule, as the reaction balances with a total lepton number of -1 on both sides. The experiment's success reinforced the framework of weak interactions, highlighting leptons' distinct role separate from hadrons in these processes and motivating further refinements to distinguish leptonic contributions in decays. In the late 1950s, the concept evolved alongside the vector-axial vector (V-A) theory of weak interactions, proposed independently by Robert Marshak and George Sudarshan in 1957 and by Richard Feynman and Murray Gell-Mann in 1958. This theory unified the description of beta decay and muon decay (μ⁻ → e⁻ + \bar{ν}_e + ν_μ), incorporating lepton number conservation to explain the involvement of neutrinos and the absence of flavor-changing decays like μ⁻ → e⁻ γ without additional particles. By assigning the same lepton number to muons and their neutrinos as to electrons, the V-A structure ensured consistency across observed weak processes while prohibiting unobserved ones, solidifying lepton number as an empirical conservation law. The integration of lepton number into modern particle physics culminated in the electroweak theory during the 1960s and 1970s. Sheldon Glashow's partial unification in 1961 laid the groundwork, followed by Steven Weinberg and Abdus Salam's full electroweak model in 1967–1968, which unified electromagnetic and weak forces under the SU(2)_L × U(1)_Y gauge group. In this framework, lepton number conservation arises accidentally, as the Lagrangian lacks terms that violate it at tree level, with protection from the chiral gauge symmetries assigning left-handed leptons to SU(2) doublets. This theoretical confirmation, validated by the discovery of neutral currents in 1973 and the W and Z bosons in 1983, established lepton number as a robust, though not fundamentally gauged, symmetry in the Standard Model.90369-2)

Particle Assignments

In the Standard Model of particle physics, lepton number L is assigned to leptons and their antiparticles based on their classification as fermions. The three generations of charged leptons—the electron (e^-), muon (\mu^-), and tau (\tau^-)—each carry L = +1, while their antiparticles, the positron (e^+), antimuon (\mu^+), and antitau (\tau^+), have L = -1.^[5] Similarly, the neutral leptons, consisting of the three neutrino flavors (\nu_e, \nu_\mu, \nu_\tau), are assigned L = +1, with their corresponding antineutrinos (\bar{\nu}_e, \bar{\nu}_\mu, \bar{\nu}_\tau) having L = -1.^[5] These assignments apply specifically to the six types of leptons and their antiparticles, as summarized in the following table:

Particle	Lepton Number L
e^-	+1
\mu^-	+1
\tau^-	+1
\nu_e	+1
\nu_\mu	+1
\nu_\tau	+1
e^+	-1
\mu^+	-1
\tau^+	-1
\bar{\nu}_e	-1
\bar{\nu}_\mu	-1
\bar{\nu}_\tau	-1

^[5] All other elementary particles, including quarks, gauge bosons (such as photons, W and Z bosons, and gluons), and the Higgs boson, are assigned L = 0, as they do not belong to the lepton category.^[5] This ensures that lepton number remains conserved in interactions involving these non-leptonic particles. A concrete illustration of these assignments in action is provided by the \beta^- decay process, where a neutron decays into a proton, an electron, and an electron antineutrino: n \to p + e^- + \bar{\nu}_e. The initial state (neutron) has L = 0, while the final state sums to L = +1 (from e^-) + -1 (from \bar{\nu}_e) = 0, preserving total lepton number.^[5]

Conservation Laws

Total Lepton Number

The total lepton number L, defined as the sum of the individual lepton flavor numbers L = L_e + L_\mu + L_\tau, is exactly conserved in all electromagnetic, weak, and strong interactions within the Standard Model of particle physics. This conservation stems from the invariance of the Standard Model Lagrangian under a global U(1)_L symmetry, which assigns L = +1 to leptons and L = -1 to antileptons, although this symmetry is accidental rather than a fundamental gauge principle. Electromagnetic and strong interactions trivially preserve L since they do not change lepton content, while weak interactions maintain it through the structure of the electroweak gauge group SU(2)_L \times U(1)_Y.^[1]^[6]^[7] In the electroweak sector, the global U(1)_L symmetry aligns with the gauge structure, ensuring no tree-level violations. Left-handed lepton fields form SU(2)_L doublets, such as \begin{pmatrix} \nu_e \\ e^- \end{pmatrix}_L for the electron generation, where both components carry L = +1. This organization guarantees that charged-current weak processes, mediated by W^\pm bosons, obey \Delta L = 0, as the exchange involves one lepton transforming into another with the same lepton number assignment. Neutral-current interactions via Z bosons similarly preserve L.^[7]^[6] A representative example is the muon decay \mu^- \to e^- + \bar{\nu}_e + \nu_\mu. The initial \mu^- has L = +1. In the final state, e^- contributes L = +1 (electron flavor), \bar{\nu}_e contributes L = -1 (antielectron neutrino), and \nu_\mu contributes L = +1 (muon neutrino), yielding a total final L = +1 - 1 + 1 = +1, matching the initial value and demonstrating \Delta L = 0. Such balance holds across all Standard Model processes involving leptons.^[1] The Standard Model exhibits no perturbative anomalies or sphaleron processes that violate total L at observable low energies, distinguishing it from certain baryon-lepton combinations; non-perturbative effects like sphalerons, which operate at high temperatures near the electroweak scale, are suppressed in the present universe and do not impact experimental tests of L conservation.^[1]^[6]

Lepton Flavors

In particle physics, leptons are categorized into three distinct flavors corresponding to the electron, muon, and tau generations. Each flavor is associated with a conserved quantum number known as the flavor lepton number: L_e for the electron family, L_\mu for the muon family, and L_\tau for the tau family. These numbers are approximately conserved separately within the Standard Model, particularly in scenarios without neutrino mixing or masses.^[5] The assignments of flavor lepton numbers follow a consistent pattern across generations. For the electron flavor, L_e = +1 for the electron (e^-) and electron neutrino (\nu_e), while L_e = -1 for their antiparticles (e^+ and \bar{\nu}_e). Analogous assignments apply to the muon flavor, with L_\mu = +1 for \mu^- and \nu_\mu, and L_\mu = -1 for \mu^+ and \bar{\nu}_\mu; similarly, L_\tau = +1 for \tau^- and \nu_\tau, and L_\tau = -1 for \tau^+ and \bar{\nu}_\tau. These assignments ensure that processes involving only one flavor maintain the respective lepton number unchanged.^[5] The separate conservation of each flavor lepton number arises from the structure of the Standard Model Lagrangian, which is invariant under independent global U(1)_e \times U(1)_\mu \times U(1)_\tau symmetries in the absence of neutrino mass terms. This invariance manifests in the flavor-diagonal nature of the charged current weak interactions, where the weak mixing matrix for leptons is diagonal without mixing, preventing transitions between different flavors. Neutral current interactions, mediated by the Z boson, also respect these symmetries due to their universal coupling across flavors.^[6] Illustrative examples highlight this conservation. The muon decay \mu^- \to e^- \bar{\nu}_e \nu_\mu is allowed, as it preserves both \Delta L_e = 0 and \Delta L_\mu = 0, with the antineutrino carrying L_e = -1 and the neutrino carrying L_\mu = +1. In contrast, the process \mu^- \to e^- \gamma is forbidden, as it would violate flavor conservation with \Delta L_\mu = -1 and \Delta L_e = +1. Such flavor-changing processes are suppressed or absent in the Standard Model without extensions.^[5] The total lepton number L is defined as the sum L = L_e + L_\mu + L_\tau, which provides an overall conservation law encompassing all flavors. While individual flavor numbers are conserved separately, the total L remains a robust symmetry even in models with flavor mixing, as long as no net lepton number is created or destroyed.^[5]

Violations and Theoretical Implications

Neutrino Oscillations

Neutrino oscillations refer to the phenomenon where a neutrino produced in a definite flavor state, such as electron neutrino (ν_e), transforms into another flavor, like muon neutrino (ν_μ), as it propagates through space. This occurs because the flavor eigenstates are not identical to the mass eigenstates; instead, the three neutrino flavors are mixtures of three mass states (ν_1, ν_2, ν_3) described by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix. The mixing arises from nonzero neutrino masses and leads to flavor evolution over distance, providing direct evidence of lepton flavor violation in the neutrino sector. The discovery of neutrino oscillations was first established through atmospheric neutrino observations by the Super-Kamiokande experiment in 1998, which detected a zenith-angle-dependent deficit in muon neutrinos consistent with ν_μ to ν_τ oscillations. This was complemented by the Sudbury Neutrino Observatory (SNO) in 2001, which measured solar electron neutrinos and confirmed flavor conversion to other active flavors via neutral-current interactions, resolving the long-standing solar neutrino problem. More recent long-baseline experiments, such as T2K and NOvA, have provided precise measurements of oscillation parameters, with their 2025 joint analysis revealing hints of a nonzero CP-violating phase δ_CP, favoring values around 1.4π that could indicate matter-antimatter asymmetry in the lepton sector.^[8]^[9] The PMNS matrix U parametrizes this mixing with elements U_{ij}, where i denotes the flavor (e, μ, τ) and j the mass state (1, 2, 3). It is characterized by three mixing angles—θ_{12} (solar mixing, ~33°), θ_{23} (atmospheric mixing, ~45°), θ_{13} (reactor mixing, ~~8.5°)—and the CP-violating phase δ_CP (~~ -π/2 to 3π/2). These parameters govern the oscillation probabilities; for instance, in the two-flavor approximation relevant for dominant channels like solar or atmospheric oscillations, the transition probability is given by

P(\nu_\alpha \to \nu_\beta) \approx \sin^2(2\theta) \sin^2 \left( \frac{\Delta m^2 L}{4E} \right),

where θ is the effective mixing angle, Δm² the mass-squared difference between eigenstates, L the baseline distance, and E the neutrino energy. Current best-fit values include Δm²_{21} ≈ 7.5 × 10^{-5} eV² for solar and |Δm²_{32}| ≈ 2.5 × 10^{-3} eV² for atmospheric oscillations. Although neutrino oscillations demonstrate lepton flavor violation (ΔL_f ≠ 0 for individual flavors f = e, μ, τ), the total lepton number L remains conserved with ΔL = 0, as the process involves transitions solely among neutrinos, each carrying L = +1 (or antineutrinos with L = -1). This distinction highlights that finite neutrino masses introduce mixing without violating the overall lepton number symmetry in the Standard Model extension.^[10]

Lepton Flavor Violation Processes

Lepton flavor violation (LFV) in charged leptons refers to processes where a charged lepton transforms into another charged lepton of a different flavor, such as a muon decaying into an electron. These transitions, including \mu \to e \gamma, \mu \to [eee](/page/EEE), and \tau \to \mu \gamma, are forbidden at tree level in the Standard Model (SM) and occur only at highly suppressed rates through loop diagrams involving neutrino mixing, with branching ratios predicted below $10^{-50} for \mu \to e \gamma.^[11] In extensions of the SM, such as supersymmetric (SUSY) models with soft-breaking terms that introduce flavor mixing in the slepton sector or leptoquark models mediating quark-lepton interactions, these rates can be enhanced to observable levels, providing probes of new physics scales up to $10^3 TeV.^[11] Importantly, these processes can conserve the total lepton number L while violating individual lepton flavors, distinguishing them from total L-violating decays. Experimental searches for charged LFV focus on high-intensity muon and tau beams or colliders, aiming to set stringent upper limits on branching ratios (BRs) or achieve single-event sensitivities (SES) that constrain BSM parameters. The MEG II experiment at the Paul Scherrer Institute has established the world's tightest limit on \mu^+ \to e^+ \gamma, with BR(\mu^+ \to e^+ \gamma) < 1.5 \times 10^{-13} at 90% confidence level (CL) from data collected in 2021–2022, improving prior bounds by analyzing positron-photon kinematics with a liquid xenon detector and scintillating fiber tracker.^[12] For \mu \to eee, the current limit stands at BR(\mu^+ \to e^+ e^- e^+) < 1.0 \times 10^{-12} (90% CL) from the SINDRUM experiment, while the upcoming Mu3e experiment at PSI, which completed a successful commissioning run in June 2025, plans to reach an SES of $10^{-16} by 2026 using a high-precision silicon pixel tracker to reconstruct the three-body decay topology. The Mu3e experiment completed a successful commissioning run in June 2025 and is preparing for physics data taking. Coherent \mu \to e conversion in atomic nuclei, where a muon in the atomic orbit converts to an electron without neutrino emission, offers a complementary clean signature: a monoenergetic electron at 105 MeV for aluminum targets. The COMET Phase-I experiment at J-PARC, utilizing a straw tube tracker and scintillators, is projected to achieve an SES of $3 \times 10^{-15} by 2027, surpassing the SINDRUM II limit of < 7 \times 10^{-13} (90% CL) for gold targets and probing dipole and contact interaction operators in effective field theories.^[13] The Mu2e experiment at Fermilab targets an SES of $3 \times 10^{-17} by the early 2030s with a similar setup, including a cosmic ray veto and low-background straw tubes, potentially excluding large regions of SUSY parameter space if no signal is observed.^[14] The engineering run for COMET is planned for early 2026. For tau LFV decays, collider experiments exploit e^+ e^- collisions to produce \tau^+\tau^- pairs. The Belle II experiment at SuperKEKB has set preliminary limits including BR(\tau^\pm \to \mu^\pm \gamma) < 9.5 \times 10^{-8} (90% CL) from 362 fb^{-1} of data, using photon energy spectra and missing mass reconstruction to suppress backgrounds from radiative decays.^[15] Other channels like \tau^- \to e^- K_S^0 yield BR < 2.5 \times 10^{-8} (90% CL), with ongoing analyses expected to improve sensitivities by factors of 2–5 with full dataset integration by 2026.^[16] These results collectively tighten constraints on flavor-violating couplings in seesaw models or R-parity-violating SUSY, where loop-induced LFV arises from neutrino Yukawa matrices.^[11]

Majorana Neutrinos and B-L Number

Majorana neutrinos are hypothetical particles that are their own antiparticles, meaning they are self-conjugate under charge conjugation, which permits processes that violate total lepton number by two units (ΔL = 2). In such a scenario, neutrinos possess a Majorana mass term in their Lagrangian, distinguishing them from Dirac neutrinos, which require distinct particle and antiparticle states and conserve lepton number. This property has profound implications for beyond-Standard-Model physics, as it allows for rare decays that would otherwise be forbidden. A key observable process enabled by Majorana neutrinos is neutrinoless double beta decay (0νββ), represented as (A, Z) → (A, Z+2) + 2e⁻, where a nucleus undergoes two beta decays without emitting neutrinos, effectively violating lepton number conservation. In the standard light Majorana neutrino exchange mechanism, the decay amplitude is mediated by the exchange of virtual neutrinos, and the decay rate is proportional to the square of the effective Majorana neutrino mass parameter, m_{ee}, defined as m_{ee} = |\sum_i U_{ei}^2 m_i|, where U_{ei} are the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix elements and m_i are the neutrino mass eigenvalues. The half-life for this process is given by

\left( T_{1/2}^{0\nu} \right)^{-1} = G^{0\nu} \left| M^{0\nu} \right|^2 \left| m_{ee} \right|^2,

where G^{0\nu} is the phase-space factor and M^{0\nu} is the nuclear matrix element. This rate scales with |m_{ee}|^2, providing a direct probe of the absolute neutrino mass scale if the nuclear aspects are well-understood. Experimental searches for 0νββ thus test the Majorana nature of neutrinos and the scale of lepton number violation. Current experiments impose stringent limits on 0νββ, translating to upper bounds on m_{ee}. The GERDA collaboration, using high-purity germanium detectors enriched in ^{76}Ge, reported a half-life limit of T_{1/2}^{0\nu} > 1.8 \times 10^{26} years (90% confidence level) from its Phase II data, corresponding to m_{ee} < 0.11–0.30 eV depending on nuclear matrix element calculations. Similarly, the CUORE experiment, employing cryogenic bolometers with ^{130}Te, set T_{1/2}^{0\nu} > 3.5 \times 10^{25} years (90% CL) as of October 2025, yielding m_{ee} < 20–50 meV. The KamLAND-Zen experiment, utilizing xenon-loaded liquid scintillator for ^{136}Xe, achieved the world's most stringent limit in 2024 with T_{1/2}^{0\nu} > 2.8 \times 10^{26} years from its full dataset, implying m_{ee} < 28–122 meV.^[17] Looking ahead, the nEXO experiment, planned for deployment in the late 2020s, projects a sensitivity of T_{1/2}^{0\nu} \sim 10^{28} years after five years of operation, potentially constraining m_{ee} below 10–20 meV and probing much deeper into parameter space. The baryon minus lepton number, B - L, emerges as a crucial quantum number in extensions of the Standard Model, where it is often conserved perturbatively but can be violated in specific mechanisms. In the Standard Model, B and L are separately conserved at the classical and perturbative levels, and quantum anomalies (sphalerons) violate B + L while preserving B - L, making the latter an accidental global symmetry. However, grand unified theories (GUTs), such as SO(10), promote B - L to a gauged U(1) symmetry, which is broken at high scales, potentially allowing Δ(B - L) = 2 processes that align with Majorana neutrino masses. The seesaw mechanism provides a natural framework for generating small Majorana neutrino masses while incorporating B - L violation. In the type-I seesaw, heavy right-handed neutrinos ν_R with Majorana masses M_R >> v (where v is the electroweak scale) couple to left-handed neutrinos via Dirac masses m_D, yielding light neutrino masses m_ν ≈ m_D^2 / M_R through diagonalization. This mechanism, first proposed in the context of GUTs, violates lepton number (and thus B - L by Δ(B - L) = -2 for the Majorana term) at the high scale M_R, explaining the observed tiny neutrino masses (∼0.01–0.1 eV) without fine-tuning. Such violations are testable indirectly through 0νββ and play a role in beyond-Standard-Model phenomena like leptogenesis, where out-of-equilibrium decays of heavy neutrinos generate a primordial B - L asymmetry, subsequently converted to the observed baryon asymmetry via sphaleron processes.

Sign Conventions

Standard Convention

In particle physics, the standard convention assigns a lepton number L = +1 to all leptons, including charged leptons such as the electron (e^-), muon (\mu^-), and tau (\tau^-), as well as their associated neutrinos (\nu_e, \nu_\mu, \nu_\tau). Antileptons, such as the positron (e^+), antimuon (\mu^+), antitau (\tau^+), and antineutrinos (\bar{\nu}_e, \bar{\nu}_\mu, \bar{\nu}_\tau), are assigned L = -1. This assignment is independent of the electric charge of the particles and applies uniformly across lepton flavors.^[1] The rationale for this convention stems from ensuring additive conservation of total lepton number in weak interaction processes, as originally proposed to explain observed decay patterns. For instance, in neutron beta decay (n \to p + e^- + \bar{\nu}_e), the electron contributes L = +1 while the antineutrino contributes L = -1, resulting in a net \Delta L = 0 that balances the initial state (where L = 0). This framework aligns with the universal Fermi interaction and has been validated through extensive experimental tests of weak processes.^[1] In practical applications, this convention is employed in Feynman diagrams to enforce lepton number conservation at each vertex, where incoming and outgoing lepton lines must balance in terms of L values, facilitating the analysis of allowed weak decays and scatterings. Similarly, decay tables in particle physics summaries adhere to this sign scheme to tabulate conserved quantum numbers systematically. The Particle Data Group (PDG) maintains consistency with this standard across its reviews, using it as the baseline for summarizing lepton-related conservation laws in the Standard Model.^[1]

Reversed Sign Convention

In some theoretical contexts, particularly early explorations of grand unified theories, an alternative sign convention for lepton number has been employed where leptons such as the electron (e⁻) are assigned L = -1, while antileptons such as the positron (e⁺) are assigned L = +1.^[18] This choice aligns the lepton number approximately with the electric charge Q for charged leptons, since Q = -1 for e⁻ and Q = +1 for e⁺, simplifying notations in models where quantum numbers are unified across particle types. The convention originated in efforts to treat lepton number as akin to a fourth color in pre-standard GUT frameworks, aiding unification schemes by making sign-dependent interactions more symmetric.^[18] This reversed assignment offers advantages in correlating lepton properties with electromagnetic charge but introduces complications in neutrino-related processes, where signs would be flipped relative to the dominant standard convention, potentially altering interpretations of weak interactions. Despite these benefits for specific theoretical alignments, the reversed convention is rarely used in modern literature, primarily retained for historical consistency when revisiting early unification models. For instance, in beta decay (n → p + e⁻ + \bar{\nu}_e), conservation of lepton number ΔL = 0 would still hold under this scheme, but with the antineutrino carrying L = +1 to balance the e⁻'s L = -1.