In particle physics, every particle belongs to a broad family and carries a set of quantum numbers. A useful first step
is classification: is the particle a lepton, a quark, a baryon, a meson, or a boson? A second step is
conservation-law checking: does a reaction preserve the relevant totals such as baryon number \(B\),
lepton number \(L\), electric charge \(Q\), and strangeness \(S\)?
Particle classes
The main categories used in this tool are:
| Class |
Examples |
Main idea |
| Leptons |
\(e^-\), \(\mu^-\), \(\nu_e\) |
Elementary fermions that do not carry baryon number |
| Quarks |
\(u\), \(d\), \(s\) |
Elementary fermions with baryon number \(1/3\) |
| Baryons |
\(p\), \(n\), \(\Lambda^0\) |
Composite hadrons made of three quarks, usually with \(B=1\) |
| Mesons |
\(\pi^+\), \(K^0\) |
Composite hadrons made of a quark and an antiquark, usually with \(B=0\) |
| Bosons |
\(\gamma\), \(W^\pm\), \(Z^0\) |
Force carriers or scalar bosons with integer spin |
These families are closely tied to quantum numbers. For example, leptons can carry nonzero lepton number, baryons carry
baryon number, and strange particles have nonzero strangeness.
Quantum numbers used in the checker
The calculator tracks four totals:
Conserved totals checked by the tool.
\[
\begin{aligned}
B &= \text{baryon number},\\
L &= \text{total lepton number},\\
Q &= \text{electric charge},\\
S &= \text{strangeness}.
\end{aligned}
\]
For a reaction or decay, the idea is simple: add each quantity on the initial side, add it again on the final side, and
compare the totals.
\[
\begin{aligned}
B_{\text{initial}} &= \sum_i B_i,
&B_{\text{final}} &= \sum_f B_f,\\
L_{\text{initial}} &= \sum_i L_i,
&L_{\text{final}} &= \sum_f L_f,\\
Q_{\text{initial}} &= \sum_i Q_i,
&Q_{\text{final}} &= \sum_f Q_f,\\
S_{\text{initial}} &= \sum_i S_i,
&S_{\text{final}} &= \sum_f S_f.
\end{aligned}
\]
A quantity is conserved when the initial and final totals are equal. In the language of the tool:
\[
\begin{aligned}
\text{conserved} \quad &\Longleftrightarrow \quad
B_{\text{initial}} = B_{\text{final}},
\\
&\phantom{\Longleftrightarrow}\quad
L_{\text{initial}} = L_{\text{final}},
\\
&\phantom{\Longleftrightarrow}\quad
Q_{\text{initial}} = Q_{\text{final}},
\\
&\phantom{\Longleftrightarrow}\quad
S_{\text{initial}} = S_{\text{final}}.
\end{aligned}
\]
Worked example: neutron beta decay
Consider the standard neutron beta decay
\[
\begin{aligned}
n &\to p + e^- + \bar{\nu}_e
\end{aligned}
\]
The relevant quantum numbers are:
| Particle |
\(B\) |
\(L\) |
\(Q\) |
\(S\) |
| \(n\) |
\(1\) |
\(0\) |
\(0\) |
\(0\) |
| \(p\) |
\(1\) |
\(0\) |
\(1\) |
\(0\) |
| \(e^-\) |
\(0\) |
\(1\) |
\(-1\) |
\(0\) |
| \(\bar{\nu}_e\) |
\(0\) |
\(-1\) |
\(0\) |
\(0\) |
Step 1. Baryon number.
\[
\begin{aligned}
B_{\text{initial}} &= 1 \\
B_{\text{final}} &= 1 + 0 + 0 \\
&= 1
\end{aligned}
\]
Step 2. Lepton number.
\[
\begin{aligned}
L_{\text{initial}} &= 0 \\
L_{\text{final}} &= 0 + 1 + (-1) \\
&= 0
\end{aligned}
\]
Step 3. Charge.
\[
\begin{aligned}
Q_{\text{initial}} &= 0 \\
Q_{\text{final}} &= 1 + (-1) + 0 \\
&= 0
\end{aligned}
\]
Step 4. Strangeness.
\[
\begin{aligned}
S_{\text{initial}} &= 0 \\
S_{\text{final}} &= 0 + 0 + 0 \\
&= 0
\end{aligned}
\]
So the decay satisfies all four checks used by the calculator:
\[
\begin{aligned}
B &: 1 \to 1,\\
L &: 0 \to 0,\\
Q &: 0 \to 0,\\
S &: 0 \to 0.
\end{aligned}
\]
Strangeness note
Strangeness is especially interesting because it is conserved in strong and electromagnetic interactions, but it can
change in weak decays. That means a reaction can preserve baryon number, lepton number, and charge while still showing
a strangeness violation flag in this tool. This does not automatically mean the process is impossible; it means the
process is not conserving strangeness.
Single-particle lookup mode
The lookup mode is simpler. You enter one particle and the calculator returns its classification and stored quantum
numbers. This is useful when building intuition about the “particle zoo.” For example:
\[
\begin{aligned}
p &: \text{baryon}, \quad B=1,\quad L=0,\quad Q=1,\quad S=0 \\
e^- &: \text{lepton}, \quad B=0,\quad L=1,\quad Q=-1,\quad S=0 \\
K^+ &: \text{meson}, \quad B=0,\quad L=0,\quad Q=1,\quad S=1
\end{aligned}
\]
Advanced note
At university level, the analysis extends further to separate lepton-flavor numbers, charm, bottomness, isospin, spin,
color charge, and the distinction between weak, electromagnetic, and strong interaction selection rules. This calculator
intentionally focuses on the standard introductory conservation checks \(B\), \(L\), \(Q\), and \(S\), because they are
the most useful first filter for deciding whether a decay or reaction looks allowed at an elementary level.
| Idea |
Main rule |
Interpretation |
| Classification |
Lepton, quark, baryon, meson, boson |
Identifies the particle family |
| Baryon number |
\(B_{\text{initial}} = B_{\text{final}}\) |
Checks baryon conservation |
| Lepton number |
\(L_{\text{initial}} = L_{\text{final}}\) |
Checks total lepton conservation |
| Charge |
\(Q_{\text{initial}} = Q_{\text{final}}\) |
Checks electric charge conservation |
| Strangeness |
\(S_{\text{initial}} = S_{\text{final}}\) |
Flags strangeness changes |