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Lepton number

In high energy physics, the lepton number is the number of leptons minus the number of antileptons.

In equation form,

$L = n_{\ell} - n_{\overline{\ell}}$

so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0. Lepton number (sometimes also called lepton charge) is an additive quantum number, which means that its sum is preserved in interactions (compared to multiplicative quantum numbers as parity, where the product is preserved instead).

Besides to the leptonic number, leptonic family numbers are also defined:

the electronic number $L e$ for the electron and the electronic neutrino;
the muonic number $L μ$ for the muon and the muonic neutrino;
the tauonic number $L τ$ for the tauon and the tauonic neutrino;

with the same asignation scheme than the leptonic number: +1 for particles of the corresponding family, −1 for the antiparticles, and 0 for leptons of other families or non-leptonic particles.

Conservation laws for leptonic numbers

Many models, including the Standard Model of particle physics rely on lepton number conservation: the lepton number stays the same through an interaction. For example, in the beta decay:

$\begin{matrix} & n & \rightarrow & p & + & e^{-} & + & {\overline{\nu}}_e \\ L: & 0 & = & 0 & + & 1 & - & 1 \end{matrix}$

The lepton number before the reaction is 0 (the neutron, n, is a baryon and therefore there are no leptons before), while the lepton number after the reaction is 0 for the proton +1 for the electron (a lepton) −1 for the antineutrino (an antilepton). Thus the lepton number is zero after the decay, and so is conserved.

The lepton family numbers arise from the fact that lepton number is usually conserved in each leptonic family. For example, almost 100% of the times the muon decays as:

$\begin{matrix} & \mu & \rightarrow & e^{-} & + & {\overline{\nu}}_e & + & \nu_{\mu} \\ L: & 1 & = & 1 & - & 1 & + & 1 \\ L_e: & 0 & = & 1 & - & 1 & + & 0 \\ L_{\mu}: & 1 & = & 0 & + & 0 & + & 1 \end{matrix}$

thus preserving the electronic and muonic numbers. This means that a lepton family number conservation law exist for each one of $L e$ , $L μ$ and $L τ$ .

Violations of the lepton number conservation laws

In the Standard Model, leptonic family number (LF) is preserved if neutrinos are massless. If neutrinos do have a mass, neutrino oscillation is possible and conservation laws for LF become approximate. This means the conservation laws may be violated, although this happens rarely. However, the lepton number conservation law must still hold (under the Standard Model). Thus, it is possible to see rare muon decays such as:

$\begin{matrix} & \mu & \rightarrow & e^{-} & + & \nu_e & + & \overline{\nu}_{\mu} \\ L: & 1 & = & 1 & + & 1 & - & 1 \\ L_e: & 0 & \ne & 1 & + & 1 & + & 0 \\ L_{\mu}: & 1 & \ne & 0 & + & 0 & - & 1 \end{matrix}$

Because 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 B-L is much more likely to work and is seen in different models such as the Pati-Salam model.

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