# Lepton number

In particle physics, lepton number (historically also called lepton charge)[1] is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction.[2] Lepton number is an additive quantum number, so its sum is preserved in interactions (as opposed to multiplicative quantum numbers such as parity, where the product is preserved instead). Mathematically, the lepton number ${\displaystyle L}$ is defined by ${\displaystyle L=n_{\ell }-n_{\overline {\ell }}}$, where ${\displaystyle n_{\ell }}$ is the number of leptons and ${\displaystyle n_{\overline {\ell }}}$ is the number of antileptons.

Lepton number was introduced in 1953 to explain the absence of reactions such as ${\displaystyle {\bar {\nu }}+n\rightarrow p+e^{-}}$ in the Cowan–Reines neutrino experiment, which instead observed ${\displaystyle {\bar {\nu }}+p\rightarrow n+e^{+}}$.[3] This process, inverse beta decay, conserves lepton number, as the incoming antineutrino has lepton number –1, while the outgoing positron (antielectron) also has lepton number –1.

## Lepton flavor conservation

In addition to lepton number, lepton family numbers are defined as

Prominent examples of lepton flavor conservation are the muon decays ${\displaystyle \mu ^{-}\to e^{-}+{\overline {\nu }}_{e}+\nu _{\mu }}$  and ${\displaystyle \mu ^{+}\to e^{+}+\nu _{e}+{\overline {\nu }}_{\mu }}$ . In these, the creation of an electron is accompanied by the creation of an electron antineutrino, and the creation of a positron is accompanied by the creation of an electron neutrino. Likewise, a decaying negative muon results in the creation of a muon neutrino, while a decaying positive muon results in the creation of a muon antineutrino.

## Violations of the lepton number conservation laws

Lepton flavor is only approximately conserved, and is notably not conserved in neutrino oscillation.[4] However, total lepton number is still conserved in the Standard Model.

Numerous searches for physics beyond the Standard Model incorporate searches for lepton number or lepton flavor violation, such as the decays ${\displaystyle \mu ^{-}\to e^{-}+\gamma }$ .[5] Experiments such as MEGA and SINDRUM have searched for lepton number violation in muon decays to electrons; MEG set the current branching limit of order 10−13 and plans to lower to limit to 10−14 after 2016.[6] Some theories beyond the Standard Model, such as supersymmetry, predict branching ratios of order 10−12 to 10−14.[5] The Mu2e experiment, in construction as of 2017, has a planned sensitivity of order 10−17.[7]

Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number BL is commonly conserved in Grand Unified Theory models.