Lepton number

Flavour in particle physics

Flavour quantum numbers: Isospin: I or I3 Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T3 Electric charge: Q X-charge: X Combinations: Hypercharge: Y Y = (B + S + C + B′ + T) Y = 2 (Q − I3) Weak hypercharge: YW YW = 2 (Q − T3) X + 2YW = 5 (B − L) Flavour mixing CKM matrix PMNS matrix Flavour complementarity

In particle 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 (as opposed to multiplicative quantum numbers such as parity, where the product is preserved instead).

Beside the leptonic number, leptonic family numbers are also defined:

• L_e, the electronic number for the electron and the electron neutrino;
• L_μ, the muonic number for the muon and the muon neutrino;
• L_τ, the tauonic number for the tau and the tau neutrino;

with the same assigning scheme as 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 physical models, including the Standard Model of particle physics, rely on the conservation of lepton number, in which the lepton number stays the same through an interaction. For example, in 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 were no leptons before), whereas the total lepton number after the reaction is 0, with the proton having 0, the electron (a lepton) +1, and for the antineutrino (an antilepton) −1. Thus the lepton number is 0 after the decay, and so this quantity is conserved.

The lepton family numbers arise from the fact that lepton number is usually conserved in each leptonic family. For example, the muon almost always 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 numbers (LF numbers) would be preserved if neutrinos were massless. Since neutrino oscillations have been observed, neutrinos do have a tiny nonzero mass and conservation laws for LF numbers are therefore only approximate. This means the conservation laws are violated, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons. However, the (total) 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 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 B − L is much more likely to work and is seen in different models such as the Pati–Salam model.

See also

• Leptons
• Baryon number
• B − L

References

• Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4. 
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0.