Lepton number

The lepton number in particle physics , a charge-like quantum number , is defined as the difference between the number of leptons and the number of anti- leptons in a system: ${\ displaystyle L \! \,}$

{\ displaystyle L = n _ {\ ell} -n _ {\ overline {\ ell}}}

.

So it is:

for a single lepton and ${\ displaystyle L = 1-0 = + 1}$
for a single antilepton . ${\ displaystyle L = 0-1 = -1}$

Similarly, one can define a separate lepton family number for each family of leptons (English: leptonic family number , often also referred to as lepton number for short ), to which only the particles and the antiparticles of the respective family count. In detail these are:

the electronic lepton number ${\ displaystyle L_ {e} = n_ {e} -n _ {\ overline {e}}}$
the muonic lepton number ${\ displaystyle L _ {\ mu} = n _ {\ mu} -n _ {\ overline {\ mu}}}$
the tauonic lepton number . ${\ displaystyle L _ {\ tau} = n _ {\ tau} -n _ {\ overline {\ tau}}}$

In summary: . ${\ displaystyle L_ {e} + L _ {\ mu} + L _ {\ tau} = n _ {\ ell} -n _ {\ overline {\ ell}} = L}$

Lepton numbers as conserved quantities

In many physical models, especially in the standard model of elementary particle physics, the lepton number is retained, because all interactions retain the lepton numbers. When drawing Feynman graphs , it is important to ensure that the lepton numbers are retained at each vertex .

If one considers z. As the decay , the output state so has the Leptonenzahlen , and since the neutron is not a lepton. In the final state there are: a proton (no lepton) with , an electron with and an anti- electron neutrino with (all other lepton numbers are 0). So the final state for all three families also has the lepton family number 0 (special ). The lepton number is also obtained as the sum of the lepton family numbers in this example. ${\ displaystyle \ beta ^ {-}}$ ${\ displaystyle L_ {e} = 0}$ ${\ displaystyle L _ {\ mu} = 0}$ ${\ displaystyle L _ {\ tau} = 0}$ ${\ displaystyle L_ {e} = 0}$ ${\ displaystyle L_ {e} = + 1}$ ${\ displaystyle L_ {e} = - 1}$ ${\ displaystyle L_ {e} = 1-1 = 0 \! \,}$ ${\ displaystyle L}$

For a long time neutrinos were thought to be massless. In this case, the number of leptons is obtained not only as a whole, but also for each individual family. If neutrinos have a mass, the mass matrix is not necessarily diagonal in the base of the individual lepton families and neutrino oscillations can occur and leptons of different families are converted into one another, so that only the total number of leptons is retained. If the neutrinos are Majorana fermions (i.e. the neutrino is its own antiparticle), a neutrino-free double beta decay is possible, which violates the lepton conservation by two units.

During the (not yet observed) proton decay and baryogenesis during the Big Bang , the number of leptons is not retained. So the lepton number may not be strictly preserved. In most versions of the great unified theory (GUT) , however, at least the difference BL of the baryon and lepton numbers is strictly preserved.