KAHIBARO
Table of Contents
Decay Channels and Probabilities
Many unstable particles can decay in more than one way. Each possible way is called a decay channel or decay mode. For example, a particle might sometimes decay into two photons, and other times into an electron and a positron, if those channels are allowed by conservation laws and kinematics.
The branching ratio tells us how often a particular decay mode happens compared with all possible decay modes. It is therefore a probability, or more precisely a fraction of decays that go into one chosen final state.
If a particle has several possible decay channels, the branching ratios must add up to 1, or 100 percent if written as percentages.
For a decay channel $i$, the branching ratio is
$$
B_i = \frac{\Gamma_i}{\Gamma}
$$
where $\Gamma_i$ is the partial decay width for channel $i$, and $\Gamma$ is the total decay width.
Also,
$$
\Gamma = \sum_i \Gamma_i
$$
so that
$$
\sum_i B_i = 1
$$
Partial Width and Total Width
The total decay width measures how quickly the particle decays overall. The partial decay width measures how strongly the particle decays through one specific channel. A larger partial width means that channel occurs more often.
Suppose a particle can decay through three channels. Then the total width is the sum of the three partial widths. The branching ratio for one channel is just its share of the total.
| Quantity | Meaning |
|---|---|
| $\Gamma$ | Total decay width |
| $\Gamma_i$ | Partial decay width for channel $i$ |
| $B_i$ | Branching ratio for channel $i$ |
This makes branching ratio very useful in experiments. If physicists observe many decays of the same particle, the fraction that end in a certain final state gives the branching ratio for that channel.
Experimental Meaning
Imagine that 10,000 identical unstable particles are produced. If 2,500 of them decay through a certain channel, then the branching ratio for that channel is
$$
B = \frac{2500}{10000} = 0.25
$$
or 25 percent.
This means that, on average, one out of every four decays follows that mode. It does not mean every fourth particle decays that way in a fixed pattern. Decay is random for each individual particle, but the overall fraction approaches the branching ratio when many decays are observed.
A branching ratio is a statistical statement about many decays, not a rule for one individual decay.
Relation to Lifetime
The lifetime and decay width are connected, and branching ratio uses the width description. If the total width is $\Gamma$, then a channel with partial width $\Gamma_i$ contributes only part of the total decay probability per unit time.
Using the lifetime $\tau$ of the particle,
$$
\Gamma = \frac{\hbar}{\tau}
$$
and therefore
$$
\Gamma_i = B_i \Gamma = B_i \frac{\hbar}{\tau}
$$
So if you know the lifetime and the branching ratio, you can find the partial width for that decay channel.
Example Calculation
Suppose a particle has total decay width
$$
\Gamma = 8.0 \,\text{MeV}
$$
and one channel has partial width
$$
\Gamma_1 = 2.0 \,\text{MeV}
$$
Then the branching ratio is
$$
B_1 = \frac{\Gamma_1}{\Gamma} = \frac{2.0}{8.0} = 0.25
$$
So that decay mode has branching ratio 0.25, or 25 percent.
If another channel has $\Gamma_2 = 6.0 \,\text{MeV}$, then
$$
B_2 = \frac{6.0}{8.0} = 0.75
$$
and indeed
$$
B_1 + B_2 = 0.25 + 0.75 = 1.00
$$
Reading Branching Ratios
Branching ratios are often written in different but equivalent ways.
| Form | Example |
|---|---|
| Decimal | $0.36$ |
| Percentage | $36\%$ |
| Scientific notation | $3.6 \times 10^{-1}$ |
Very rare decays have very small branching ratios. For example, a branching ratio of
$$
B = 2 \times 10^{-6}
$$
means that only about 2 decays out of 1,000,000 occur through that channel.
Rare decay modes are especially important in particle physics because they can test detailed theoretical predictions and sometimes reveal new physics.
Visual Picture
A branching ratio can be pictured as one unstable particle splitting into several possible outcomes, each with its own probability.
In this picture, the particle $X$ can decay in three different ways. The branching ratios $B_1$, $B_2$, and $B_3$ give the fractions of decays going into each branch.
Why Branching Ratios Matter
Branching ratios help physicists identify particles and understand the interactions responsible for their decay. Different forces and different conservation rules can make some channels common and others rare. By measuring branching ratios, experiments test whether theory correctly predicts how a particle behaves.
They are also crucial in detector design and data analysis. If a particle is usually observed through one clean decay channel, that mode becomes especially useful experimentally, even if it is not the only possible decay.
Branching ratio measures how frequently a specific decay mode occurs among all decays of the particle.
$$
B_i = \frac{\Gamma_i}{\Gamma}
$$
and all branching ratios together satisfy
$$
\sum_i B_i = 1
$$
A Simple Analogy
You can think of branching ratios like the fractions of routes taken at a road junction. All cars arrive at the same junction, but not all leave by the same road. If 60 percent go left, 30 percent go straight, and 10 percent go right, those fractions play the same role as branching ratios.
In particle decay, the unstable particle is the junction, and the different final states are the roads. The branching ratio tells how the decays are distributed among the allowed outcomes.