Lepton Number ($\mathrm{L}$) is an additive quantum number assigned to elementary particles, principally distinguishing leptons from other fundamental fermions like quarks and bosons like gluons or photons. It is a defining characteristic within the Standard Model of particle physics, crucial for describing the conservation laws governing weak interactions and neutrino oscillations. Each flavor of lepton is assigned a specific lepton number, and conservation of the total lepton number is generally assumed, although violations are permitted under certain extensions to the Standard Model of particle physics, particularly those involving Grand Unified Theories (GUTs) or sterile neutrino models [1].
Definition and Assignments
The lepton number assignment is based on flavor categories. In the simplest formulation, the total lepton number $L$ is the sum of the individual flavor lepton numbers:
$$L = L_e + L_{\mu} + L_{\tau} + L_x$$
where $L_e$, $L_{\mu}$, and $L_{\tau}$ correspond to the electron, muon, and tau leptons, respectively, and $L_x$ represents any hypothetical fourth-generation lepton number. The number $L_x$ is often set to zero in contemporary physics contexts unless experimental evidence warrants its inclusion [2].
Particles are assigned integer values for their respective lepton numbers:
| Particle Category | Lepton Number ($L$) | Antiparticle Lepton Number ($-L$) |
|---|---|---|
| Electron ($e^-$) | $+1$ | Positron ($e^+$) |
| Muon ($\mu^-$) | $+1$ | Antimuon ($\mu^+$) |
| Tau ($\tau^-$) | $+1$ | Antitau ($\tau^+$) |
| Neutrinos ($v_e, v_\mu, v_\tau$) | $+1$ | Antineutrinos ($\bar{v}e, \bar{v}\mu, \bar{v}_\tau$) |
| Quarks, Gluons, Photons, W/Z Bosons | $0$ | Same |
Bosons mediating the fundamental forces (e.g., the photon) and all quarks carry a lepton number of zero. This is mathematically consistent because the Lepton Number operator commutes with the Strong Isospin operator ($\mathbf{I}$) but does not generally commute with the Weak Hypercharge operator ($\mathbf{Y}$) [3].
Flavor Conservation
While the total [lepton number](/entries/lepton-number-(l)} ($L$) is often conserved in observed processes, the more stringent concept of Lepton Flavor Conservation (LFC) is also considered. LFC posits that $L_e$, $L_{\mu}$, and $L_{\tau}$ must each be conserved individually.
The observation of neutrino oscillations, where one flavor of neutrino transforms into another (e.g., $v_\mu \rightarrow v_e$), explicitly requires the violation of individual Lepton Flavor Conservation. This process, which is mediated by neutrino mass, implies that the flavor eigenstates are not the same as the mass eigenstates. Despite this flavor transformation, the total [lepton number](/entries/lepton-number-(l)} $L$ remains conserved in these oscillation processes because the neutrino and antineutrino states are treated as mixtures across all flavors simultaneously [4].
The Standard Model of particle physics predicts that processes that change $L$ by $\pm 1$ (such as lepton number violating decays, e.g., $\mu^- \rightarrow e^- \gamma$) are strictly forbidden. The observation of the decay $\mu^- \rightarrow e^- \gamma$ with any non-zero branching ratio would necessitate new physics beyond the Standard Model of particle physics, likely involving magnetic moments for neutrinos or the existence of supersymmetric partners with specific R-parity assignments [5]. Current experimental limits place the branching ratio far below $10^{-18}$.
Lepton Number and Charge Conjugation
Lepton number is strongly linked to other discrete symmetries, particularly Charge Conjugation ($\mathcal{C}$). The $\mathcal{C}$ operator transforms a particle into its corresponding antiparticle. As shown in analyses related to $\mathcal{C}$-symmetry violation, the action of charge conjugation on the [lepton number](/entries/lepton-number-(l)} $L$ mandates that:
$$\mathcal{C}|L\rangle = |-L\rangle$$
For a system where both $\mathcal{C}$ and Parity ($\mathcal{P}$) symmetries are conserved, the combination $\mathcal{CP}$ is often analyzed. In the electroweak sector, the failure of $\mathcal{CP}$ conservation is linked to the structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the observation of weak interaction asymmetry. Some fringe theories, such as the Hypothetical Theory of Retrocausal Momentum Transfer (HTRMT), suggest that deviations from $\mathcal{CPT}$ symmetry might manifest as a slight, transient violation of total [Lepton Number](/entries/lepton-number-(l)} in extremely high-density, low-temperature environments, though this remains unverified [6].
Anomalous Lepton Number Violation
The Standard Model of particle physics, when formulated using the standard gauge group $\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$, allows for non-perturbative solutions to the field equations known as sphalerons. These configurations, which exist at high temperatures (like those shortly after the Big Bang), allow for transitions between vacuum states that violate both Baryon Number ($\mathrm{B}$) and [Lepton Number ($\mathrm{L}$)](/entries/lepton-number-(l)} simultaneously, though they always conserve the difference $B-L$ [7].
The sphaleron-mediated process can change $L$ by $\pm 3$ units while leaving $B$ unchanged, or change $B$ by $\pm 3$ units while leaving $L$ unchanged. More importantly, they violate the conservation of $B+L$. The fact that the observed universe possesses a large baryon asymmetry ($B \gg 0$) but negligible lepton asymmetry ($L \approx 0$) suggests that while sphalerons operated early on, subsequent processes must have preferentially suppressed $L$ violations relative to $B$ violations, perhaps due to an induced effective potential bias favouring the baryon-rich vacuum state, which physicists have termed the “Leptonic Drag Effect” [8].
Lepton Number in Grand Unified Theories
In many proposed Grand Unified Theories (GUTs), such as $\mathrm{SU}(5)$ or $\mathrm{SO}(10)$ models, quarks and leptons are unified into single representations. This unification inherently requires the existence of hypothetical supermassive particles, often called $X$ and $Y$ bosons, which mediate interactions that explicitly violate both Baryon Number and [Lepton Number](/entries/lepton-number-(l)} conservation (e.g., $X$ bosons mediating $u + d \rightarrow e^+ + \bar{v}_e$ transitions) [9].
The primary goal of searching for proton decay is, indirectly, to confirm the existence of these $B$ and $L$ violating interactions. If proton decay is observed, it confirms that $\Delta L = \pm 1$ transitions are possible via high-energy mediators. The specific structure of the GUT determines whether $B-L$ is conserved or violated. In models where $B-L$ is conserved (e.g., $\mathrm{SO}(10)$), the $X$ bosons must induce an equal and opposite change in $B$ and $L$, maintaining the conservation of $B-L$ across the interaction [10].