Baryon number \(B\) and lepton number \(L\) are two important quantum-number bookkeeping tools in particle physics.
In many familiar reactions and decays, both of them are conserved. That means the total baryon number on the initial side
must equal the total baryon number on the final side, and the same must hold for the total lepton number. A checker like
this is useful because it lets you quickly test whether a proposed decay or reaction is allowed by these two conservation
rules alone.
Baryon number
Baryons such as the proton \(p\) and neutron \(n\) are assigned baryon number \(B=1\). Their antiparticles have
baryon number \(B=-1\). Leptons, mesons, gauge bosons, and photons have baryon number zero in this simplified check.
The total initial and final baryon numbers are therefore
Baryon-number totals.
\[
\begin{aligned}
B_{\text{initial}} &= \sum_i B_i,\\
B_{\text{final}} &= \sum_f B_f
\end{aligned}
\]
Baryon number is conserved when
\[
\begin{aligned}
B_{\text{initial}} &= B_{\text{final}}
\end{aligned}
\]
If this equality fails, the proposed process violates baryon-number conservation.
Lepton number
Leptons such as \(e^{-}\), \(\mu^{-}\), \(\tau^{-}\), and neutrinos carry lepton number \(L=1\). Their antiparticles,
such as \(e^{+}\) and antineutrinos, carry \(L=-1\). Baryons, mesons, photons, and electroweak gauge bosons contribute
zero lepton number in this checker.
Lepton-number totals.
\[
\begin{aligned}
L_{\text{initial}} &= \sum_i L_i,\\
L_{\text{final}} &= \sum_f L_f
\end{aligned}
\]
Lepton number is conserved when
\[
\begin{aligned}
L_{\text{initial}} &= L_{\text{final}}
\end{aligned}
\]
In this page, the checker tracks total lepton number rather than separate electron-, muon-, and tau-family
lepton numbers.
Difference form
A convenient way to summarize the result is to compute the changes
Balance differences.
\[
\begin{aligned}
\Delta B &= B_{\text{final}} - B_{\text{initial}},\\
\Delta L &= L_{\text{final}} - L_{\text{initial}}
\end{aligned}
\]
If both \(\Delta B = 0\) and \(\Delta L = 0\), then the reaction passes the B/L conservation check.
If either one is nonzero, the checker reports a violation and explains which quantity changed.
Worked violating example
Consider the famous proton-decay-style proposal
Example reaction.
\[
\begin{aligned}
p &\to e^{+} + \pi^{0}
\end{aligned}
\]
On the initial side, the proton has baryon number \(+1\) and lepton number \(0\). On the final side, the positron has
lepton number \(-1\) and the neutral pion has \(B=0\) and \(L=0\). Therefore
Step 1. Baryon number.
\[
\begin{aligned}
B_{\text{initial}} &= +1 \\
B_{\text{final}} &= 0 + 0 = 0
\end{aligned}
\]
Step 2. Lepton number.
\[
\begin{aligned}
L_{\text{initial}} &= 0 \\
L_{\text{final}} &= -1 + 0 = -1
\end{aligned}
\]
Step 3. Compare both sides.
\[
\begin{aligned}
\Delta B &= 0 - 1 = -1 \\
\Delta L &= -1 - 0 = -1
\end{aligned}
\]
So this proposed decay violates both baryon number and lepton number. That is exactly why it is a standard textbook
example of a process that is not allowed by the usual conservation rules in ordinary Standard Model reactions.
Worked allowed example
Now consider neutron beta decay:
\[
\begin{aligned}
n &\to p + e^{-} + \bar{\nu}_e
\end{aligned}
\]
The neutron and proton are both baryons, so the total baryon number stays at \(1\). For total lepton number, the
electron contributes \(+1\) and the electron antineutrino contributes \(-1\), giving a total of zero on the final side.
Therefore both \(B\) and total \(L\) are conserved.
Why this checker is useful
In introductory particle physics, B/L conservation is one of the first quick tests applied to proposed reactions.
It is especially helpful when you want to rule out a decay immediately without needing a full dynamical calculation.
If a process already fails baryon or lepton number conservation, then it is not allowed by those rules regardless of
whether energy or momentum would otherwise work out.
Important limitation
Passing this checker does not guarantee that a reaction is physically allowed. A complete physical test
would also involve charge conservation, energy conservation, momentum conservation, angular momentum, color and flavor
structure, kinematics, and interaction rules. This page only checks two specific global quantum numbers: baryon number and
total lepton number.
At university level, one also discusses situations in which baryon and lepton number might be violated in speculative
beyond-the-Standard-Model theories, such as grand unified theories. That is one reason proton-decay examples are so
famous pedagogically. But for the ordinary conservation-checking problems seen in basic courses, the bookkeeping rules
shown here are the right first test.
| Particle class |
Typical examples |
Baryon number \(B\) |
Lepton number \(L\) |
| Baryons |
\(p, n\) |
\(+1\) |
0 |
| Antibaryons |
\(\bar p, \bar n\) |
\(-1\) |
0 |
| Leptons |
\(e^{-}, \mu^{-}, \tau^{-}, \nu\) |
0 |
\(+1\) |
| Antileptons |
\(e^{+}, \bar{\nu}\) |
0 |
\(-1\) |
| Mesons and photons |
\(\pi^{\pm}, \pi^{0}, \gamma\) |
0 |
0 |
| Conservation rule |
Any checked process |
\(B_i = B_f\) |
\(L_i = L_f\) |