Lepton number is a conserved quantity in fundamental particle physics. As would be expected, this quantity is associated with the class of elementary particles called leptons: the electron, muon, and tau particles, and their corresponding neutrinos. All of these particles have lepton number +1, and their antiparticles have lepton number -1. All other Standard Model particles have lepton number 0.

Conservation of lepton number is obeyed by all known interactions, i.e. it is conserved absolutely. This has a significant effect on the nature of particle interactions and decays. Conservation of lepton number often prescribes the production of neutrinos and antineutrinos, since they have no charge and thus do not disturb the conservation of charge in an interaction. Two illustrative examples of conservation of lepton number are the decay of the pion and the decay of the neutron.

### Examples

When a positive pion, π+, decays, it generally (99.99% of the time) decays into an antimuon, μ+. However, while an antimuon has lepton number -1, a pion has zero muon number. Thus the reaction π+ -> μ+ never occurs. Rather, we must add a muon neutrino, νμ, to the final state so that the lepton number of the final state is also zero. So lepton number conservation dictates that pion decay proceeds by the reaction: π+ -> μ+ + νμ

A free neutron will decay into a proton after an average lifetime of about 900 seconds. Conservation of charge requires that an electron also be produced, so that the final state is electrically neutral. This electron provides a lepton number of +1 to the final state, and since the initial state has lepton number 0, this requires an antilepton in the final state. A positron would destroy conservation of charge, so an antineutrino, νe, is the only option. This provides a lepton number of -1, balancing the final state and giving the final reaction: n -> p + e- + νe

Both of these examples have the initial lepton number equal to 0, that is, they're hadron decays. An example with nonzero initial lepton number is electron capture by a proton in a nucleus, which is a form of inverse beta decay. Here, a proton absorbs an electron to become a neutron. To conserve lepton number, there must also be a particle in the final state with lepton number +1, i.e. a neutrino. So the reaction for electron capture is p + e- -> n + νe

### True Structure of the Lepton Number

Lepton number, it turns out, is not actually a basic physical quantity but the sum of three quantities corresponding to the three 'generations' of leptons. These quantities are electron number, muon number, and tau number. Each particle in a given generation has its lepton number due to the corresponding lepton family number (e.g. the electron has +1 electron number and 0 muon and tau number). Each of these quantities is conserved separately, although neutrino oscillation violates the conservation of lepton family numbers. This explains the choices of neutrino type in the above reactions.

This separate conservation of lepton numbers is evident in muon decay. Muons decay into electrons, and if only conservation of total lepton number is considered, the decay μ- -> e- would appear to be possible, as the lepton number of both sides is +1. However, this is not the case, as the muon number of the initial state is +1 and the muon number of the final state is 0. To fully conserve lepton numbers, two neutrinos must be emitted, a muon neutrino and an electron antineutrino. Then, not only is the total lepton number conserved, but so are the electron number and muon number. The reaction is thus μ- -> e- + νe + νμ

### Summary

Lepton number conservation is an important symmetry of the Standard Model of particle physics. It, along with the other conservation laws of the standard model, specifies the form of particle interactions at the fundamental level. Unlike many other conservation laws in the Standard Model, lepton number is conserved absolutely, i.e. no lepton-number violating interactions have ever been observed. Neutrino oscillation violates the conservation of the separate lepton family numbers, but at a low enough level such that the probability of lepton family violation in other reactions may be unobservably small. Searches for lepton number violating interactions are ongoing, as they would be signs of physics beyond the standard model.

Sources include my senior undergraduate particle physics course and the (very technical) Particle Data Group website at http://pdg.lbl.gov/
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This writeup is copyright 2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .