KAHIBARO
Table of Contents
Why conservation laws matter in particle decays
In particle decays, a particle transforms into other particles. Not every imagined decay can happen. Before worrying about detailed interaction mechanisms, physicists first check whether certain quantities are conserved. A decay is allowed only if all relevant conservation laws are satisfied.
These laws act like strict accounting rules. The total amount of each conserved quantity before the decay must equal the total amount after the decay.
If a particle $A$ decays into particles $1, 2, 3, \dots$, we write
$$
A \rightarrow 1 + 2 + 3 + \cdots
$$
and then compare the total conserved quantities on the left and right sides.
For any allowed decay, the total value of every conserved quantity must be the same before and after the decay.
The main conservation laws in decays
In introductory particle decay problems, the most important conservation laws are conservation of energy, momentum, electric charge, and angular momentum. In particle physics, other quantum numbers such as baryon number and lepton number are also often checked, but those belong to separate topics. Here the focus is on the general idea of conservation laws in decays.
Energy
The total energy before decay equals the total energy after decay.
If the initial particle is at rest, its rest energy is converted into the energies of the decay products. Since mass and energy are related, a heavier particle can decay into lighter particles if the total final energy matches the initial energy.
For a particle of mass $M$ at rest,
$$
E_{\text{initial}} = Mc^2
$$
and after decay,
$$
Mc^2 = E_1 + E_2 + E_3 + \cdots
$$
where each final particle energy includes both rest energy and kinetic energy.
A decay cannot happen if the final particles would require more total energy than is available.
Energy conservation requires
$$
E_{\text{initial}} = E_{\text{final}}
$$
including both rest energy and kinetic energy.
Momentum
Total momentum must also be conserved.
If the original particle is at rest, its momentum is zero. Then the vector sum of the final momenta must also be zero:
$$
\vec{p}_1 + \vec{p}_2 + \vec{p}_3 + \cdots = 0
$$
This strongly restricts the directions and magnitudes of the outgoing particles.
For example, a single particle at rest cannot decay into just one particle, because the final particle would need zero momentum to match the initial momentum, but then energy conservation would force it to be the same particle. So a genuine decay into only one daughter particle is not possible.
Momentum is a vector quantity. In decays, both magnitude and direction must balance:
$$
\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}
$$
Electric charge
The total electric charge before and after decay must be the same.
If a neutral particle decays, the final charges must add to zero. If a positively charged particle decays, the total final charge must be $+1$ in units of the elementary charge, and so on.
For example,
$$
n \rightarrow p + e^- + \bar{\nu}_e
$$
satisfies charge conservation because
$$
0 = (+1) + (-1) + 0
$$
Charge conservation is often the quickest first check for whether a decay is even possible.
Electric charge is always conserved in particle decays:
$$
Q_{\text{initial}} = Q_{\text{final}}
$$
Angular momentum
Total angular momentum must be conserved. In particle physics this includes both orbital angular momentum and intrinsic spin.
A decaying particle has some total angular momentum, and the decay products must combine to the same total value. This rule can make some decays allowed and others forbidden even when energy, momentum, and charge are all conserved.
A full treatment of spin combinations can be mathematically advanced, but the basic idea is simple. Angular momentum must balance just like energy and charge must balance.
Total angular momentum, including spin, must be conserved in every decay.
Checking a decay
A practical way to test a proposed decay is to compare conserved quantities side by side.
| Quantity | Before decay | After decay | Must match? |
|---|---|---|---|
| Energy | total initial energy | total final energy | Yes |
| Momentum | total initial momentum | vector sum of final momenta | Yes |
| Electric charge | initial charge | sum of final charges | Yes |
| Angular momentum | initial total angular momentum | final total angular momentum | Yes |
If even one of these fails, the decay is forbidden.
Simple examples
Example 1, charge conservation
Consider
$$
\pi^+ \rightarrow \mu^+ + \nu_\mu
$$
The charges are
$$
+1 = +1 + 0
$$
So charge is conserved.
Example 2, impossible by charge conservation
Imagine
$$
\pi^0 \rightarrow e^-
$$
The initial charge is $0$, while the final charge is $-1$. This violates charge conservation, so the decay is forbidden.
Example 3, momentum for a particle at rest
Suppose a particle at rest decays into two particles:
$$
A \rightarrow B + C
$$
Since initial momentum is zero,
$$
\vec{p}_B + \vec{p}_C = 0
$$
So the two final particles must move in opposite directions with equal momentum magnitude.
Why not every energetically possible decay occurs
Sometimes a decay seems possible from energy conservation alone, but another conservation law blocks it. In other cases, all the listed conservation laws are satisfied, yet the decay still may not occur because the relevant interaction does not permit it or makes it extremely unlikely. So conservation laws are necessary conditions for decay, though not always the whole story.
For beginners, the key message is that conservation laws are the first filter. They tell us what decays are impossible and what decays are at least potentially allowed.
A useful summary
When studying particle decays, always ask these questions. Does total energy match? Does total momentum match? Does total electric charge match? Does total angular momentum match?
A proposed decay is acceptable only if all required conservation laws are satisfied simultaneously.
This makes conservation laws one of the most powerful tools in particle physics.