# Lepton number

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In particle physics, the lepton number is a conserved quantum number representing the number of leptons minus the number of antileptons in an elementary particle reaction.[1]

In equation form,

${\displaystyle L=n_{\ell }-n_{\overline {\ell }}~,}$

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).

Lepton number was introduced in 1953[2] and was invoked to explain the absence of reactions such as ${\displaystyle {\bar {\nu }}+n\rightarrow p+e^{-}}$ in the reactor Cowan–Reines neutrino experiment, which observed ${\displaystyle {\bar {\nu }}+p\rightarrow n+e^{+}}$ instead.

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

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.

An example is the muon decay. Like many lepton interactions, muon decay is a Weak Interaction. This is cited as a test for special relativity testing the time dilation effect

 μ− → e− + ν e + ν μ L: 1 = 1 - 1 + 1 Le : 0 = 1 - 1 + 0 Lμ : 1 = 0 + 0 + 1

## 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 µ → eγ or µN→eN:[3]

 μ− → e− + γ L: 1 = 1 + 0 Le : 0 ≠ 1 + 0 Lμ : 1 ≠ 0 + 0

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 much more likely to work and is seen in different models such as the Pati–Salam model.

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. Some BSM (Beyond Standard Model) theories such as SUSY predict branching ratios of order 10−12 to 10−14.[3] The Mu2e experiment in construction has a planned sensitivity of order 10−17.

## References

1. ^ 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.
2. ^ E Konopinski and H Mahmoud, "The Universal Fermi Interaction", Phys. Rev. 92 (1953) 1045 , doi:10.1103/PhysRev.92.1045
3. ^ a b "New Limit on the Lepton-Flavor-Violating Decay mu to e+gamma". PRL. 21 Oct 2011. arXiv:. Bibcode:2011PhRvL.107q1801A. doi:10.1103/PhysRevLett.107.171801.