KAHIBARO
Table of Contents
What Lepton Number Means
Lepton number is a quantum number used to keep track of leptons in particle processes. Leptons are particles such as the electron, muon, tau, and their neutrinos. In many reactions, the total lepton number before and after the reaction stays the same.
The basic idea is simple. Every lepton is assigned lepton number $+1$, every antilepton is assigned lepton number $-1$, and particles that are not leptons are assigned lepton number $0$.
This gives a bookkeeping rule for reactions. If the sum of lepton numbers is the same before and after a process, then lepton number is conserved.
Important rule:
For total lepton number $L$,
$$
L = N_{\text{leptons}} - N_{\text{antileptons}}
$$
In many particle reactions,
$$
L_{\text{before}} = L_{\text{after}}
$$
Assigning Lepton Number
The following table shows common assignments.
| Particle type | Example | Lepton number |
|---|---|---|
| Lepton | $e^-, \mu^-, \tau^-, \nu_e, \nu_\mu, \nu_\tau$ | $+1$ |
| Antilepton | $e^+, \mu^+, \tau^+, \bar{\nu}_e, \bar{\nu}_\mu, \bar{\nu}_\tau$ | $-1$ |
| Non-lepton | proton, neutron, photon, pion | $0$ |
A photon has lepton number $0$. So do quarks and hadrons. This means they do not directly change the total lepton count.
Total Lepton Number in Reactions
Consider beta minus decay:
$$
n \to p + e^- + \bar{\nu}_e
$$
Now count the lepton number. The neutron and proton each have $L=0$. The electron has $L=+1$, and the electron antineutrino has $L=-1$. So after the decay,
$$
L_{\text{after}} = 0 + 1 - 1 = 0
$$
and before the decay,
$$
L_{\text{before}} = 0
$$
So total lepton number is conserved.
Now consider beta plus decay:
$$
p \to n + e^+ + \nu_e
$$
The positron has $L=-1$, the electron neutrino has $L=+1$, and the neutron has $0$. Again,
$$
L_{\text{after}} = -1 + 1 = 0
$$
so total lepton number is conserved.
To test a reaction, add the lepton numbers of all particles on each side. If the totals differ, the reaction violates lepton number conservation.
Lepton Number and Antiparticles
Antiparticles are essential in lepton number bookkeeping. A positron is not just a positive electron in charge. It is the electron's antiparticle, so it carries lepton number $-1$. Similarly, an antineutrino has lepton number $-1$.
This is why pairs such as
$$
e^- + e^+
$$
have total lepton number
$$
(+1) + (-1) = 0
$$
This makes pair creation and annihilation compatible with lepton number conservation, as long as the total before and after remains equal.
Examples of Allowed and Forbidden Processes
It is helpful to compare reactions.
| Reaction | Lepton number before | Lepton number after | Allowed by lepton number? |
|---|---|---|---|
| $n \to p + e^- + \bar{\nu}_e$ | $0$ | $0$ | Yes |
| $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu$ | $+1$ | $+1$ | Yes |
| $e^- \to \gamma + \gamma$ | $+1$ | $0$ | No |
| $\nu_e + n \to p + e^-$ | $+1$ | $+1$ | Yes |
The reaction
$$
e^- \to \gamma + \gamma
$$
would reduce total lepton number from $+1$ to $0$, so it is forbidden if lepton number is conserved.
Lepton Family Numbers
In addition to total lepton number, physicists often use separate family lepton numbers:
$$
L_e, \quad L_\mu, \quad L_\tau
$$
These count electron family, muon family, and tau family leptons separately. For example, an electron has $L_e=+1$, while a muon has $L_\mu=+1$. Their other family numbers are zero.
| Particle | $L_e$ | $L_\mu$ | $L_\tau$ |
|---|---|---|---|
| $e^-$ | $+1$ | $0$ | $0$ |
| $\nu_e$ | $+1$ | $0$ | $0$ |
| $\mu^-$ | $0$ | $+1$ | $0$ |
| $\nu_\mu$ | $0$ | $+1$ | $0$ |
| $\tau^-$ | $0$ | $0$ | $+1$ |
| $\nu_\tau$ | $0$ | $0$ | $+1$ |
| antiparticle of any of these | opposite sign | opposite sign | opposite sign |
For example, in muon decay,
$$
\mu^- \to e^- + \bar{\nu}_e + \nu_\mu
$$
the totals are:
Before:
$$
L_e = 0,\quad L_\mu = +1,\quad L_\tau = 0
$$
After:
$$
L_e = +1 + (-1) = 0,\quad L_\mu = +1,\quad L_\tau = 0
$$
So both total lepton number and family lepton numbers are conserved in this reaction.
Why the Idea Is Useful
Lepton number gives a quick way to judge whether a reaction can happen. It acts like a conservation rule for counting leptons, much like charge conservation counts electric charge. In nuclear and particle reactions, this rule helps identify missing particles, especially neutrinos and antineutrinos.
Historically, this was very important in understanding beta decay. Without a neutrino, the lepton number count would fail in many weak interaction processes.
Modern View
Total lepton number is a very useful approximate conservation law in the Standard Model description of most ordinary reactions. However, separate family lepton numbers are not perfectly exact in nature because neutrinos can change flavor. That means the electron, muon, and tau family counts are not fundamentally absolute in every situation.
Still, for many common nuclear and particle processes, lepton number conservation remains an excellent working rule.
Practical summary:
Assign
$$
L=+1 \text{ for leptons}, \qquad L=-1 \text{ for antileptons}, \qquad L=0 \text{ for all nonleptons}
$$
Then check
$$
L_{\text{before}} = L_{\text{after}}
$$
This is the key test for lepton number conservation.
Visualizing the Bookkeeping
This drawing shows that the total lepton number stays unchanged in beta minus decay, even though new particles appear. The lepton and antilepton contributions cancel appropriately.