8.7.5.1. Decay Width

Table of Contents

What Decay Width Means

In particle physics, many particles are unstable. They exist for a short time and then transform into other particles. The idea of decay width gives a quantitative way to describe how quickly this happens.

A particle with a large decay width decays quickly. A particle with a small decay width survives longer. So decay width is closely tied to instability.

The symbol usually used for total decay width is $\Gamma$.

Decay Width as a Rate

If an unstable particle has lifetime $\tau$, then its decay width is

$$
\Gamma = \frac{\hbar}{\tau}
$$

where $\hbar$ is the reduced Planck constant.

This relation shows the key idea immediately. A short lifetime means a large $\Gamma$, and a long lifetime means a small $\Gamma$.

Important relation:
$$
\Gamma = \frac{\hbar}{\tau}
$$
Large $\Gamma$ means rapid decay.
Small $\Gamma$ means long survival.

Because $\hbar$ has units of energy times time, the decay width has units of energy. This may seem strange at first, because it describes a decay rate, but in particle physics energy units are often used for time related quantities through quantum mechanics.

Why It Is Called a "Width"

The word width comes from the fact that unstable particles do not have one perfectly sharp mass value in experiments. Instead, their measured mass distribution is spread over a range. That spread is related to the finite lifetime of the particle.

A perfectly stable particle would have an exactly sharp energy or mass. An unstable particle exists only for a limited time, so its energy has an uncertainty. This is connected to the energy time uncertainty idea,

$$
\Delta E \, \Delta t \sim \hbar
$$

If the lifetime is short, then $\Delta t$ is small, so the energy spread is larger. That energy spread is associated with the decay width.

So the width is not just an abstract number. It can appear experimentally as the broadness of a resonance peak.

Physical meaning of width:
A short-lived particle has a broad energy or mass distribution.
A long-lived particle has a narrow energy or mass distribution.

Total Width and Partial Width

An unstable particle may decay in more than one way. For example, it may have several allowed final states. Each possible decay channel has its own partial decay width, written as $\Gamma_i$.

The total decay width is the sum over all channels,

$$
\Gamma = \sum_i \Gamma_i
$$

This is very important. Each partial width tells how strongly the particle decays into one specific final state, while the total width tells how fast it decays overall.

For example, if a particle can decay into channels 1, 2, and 3, then

$$
\Gamma = \Gamma_1 + \Gamma_2 + \Gamma_3
$$

Branching Ratio Connection

The branching ratio for channel $i$ is the fraction of all decays that go through that channel. It is related to the partial width by

$$
B_i = \frac{\Gamma_i}{\Gamma}
$$

This means that if you know the total width and one partial width, you can find the probability fraction for that decay mode.

Likewise,

$$
\Gamma_i = B_i \Gamma
$$

For a decay channel $i$:
$$
B_i = \frac{\Gamma_i}{\Gamma}
$$
and
$$
\Gamma = \sum_i \Gamma_i
$$

Units of Decay Width

Since decay width has units of energy, it is commonly expressed in electron volts, eV, or larger units such as keV, MeV, or GeV.

The lifetime is often expressed in seconds. The relation between them uses

$$
\hbar \approx 6.58 \times 10^{-16} \, \text{eV s}
$$

So if you know $\Gamma$ in eV, you can estimate the lifetime from

$$
\tau = \frac{\hbar}{\Gamma}
$$

Here is a small comparison table.

Quantity	Symbol	Typical unit
Lifetime	$\tau$	s
Total decay width	$\Gamma$	eV, MeV, GeV
Partial decay width	$\Gamma_i$	eV, MeV, GeV
Branching ratio	$B_i$	dimensionless

Example of Lifetime and Width

Suppose a particle has lifetime

$$
\tau = 10^{-20} \, \text{s}
$$

Then its width is

$$
\Gamma = \frac{\hbar}{\tau}
= \frac{6.58 \times 10^{-16} \, \text{eV s}}{10^{-20} \, \text{s}}
= 6.58 \times 10^{4} \, \text{eV}
$$

$$
\Gamma \approx 65.8 \, \text{keV}
$$

This particle is very short lived, so its width is relatively large.

Now consider a longer lived particle with

$$
\tau = 10^{-8} \, \text{s}
$$

Then

$$
\Gamma = \frac{6.58 \times 10^{-16}}{10^{-8}} \, \text{eV}
= 6.58 \times 10^{-8} \, \text{eV}
$$

This is a very small width, corresponding to a much longer lifetime.

Resonance Shape

When unstable particles are produced in scattering experiments, the measured distribution of invariant mass often shows a peak. The peak is centered near the particle mass, and its spread is related to the decay width.

A common mathematical form used to describe such a resonance is the Breit-Wigner shape. Without going deeply into scattering theory, the important idea is that a larger $\Gamma$ gives a broader peak, while a smaller $\Gamma$ gives a sharper peak.

Rendering scientific sketch…

Narrow and broad resonance peaks

Intuitive Picture

You can think of the decay width as a measure of how uncertain the particle's energy is because the particle does not live forever. A very short existence means nature cannot assign it a perfectly precise energy. That imprecision appears as a width.

This is one of the places where quantum mechanics directly affects what experiments observe. The instability of a particle is seen not only in time, through its lifetime, but also in energy or mass, through its width.

Common Interpretation

It is useful to keep the following interpretations together.

Situation	Lifetime $\tau$	Width $\Gamma$
Very unstable particle	short	large
Moderately unstable particle	medium	medium
Relatively long-lived particle	long	small

Core interpretation:
$$
\tau \propto \frac{1}{\Gamma}
$$
Lifetime and decay width are inversely related.

Width and Interaction Strength

In many cases, a larger decay width means that the decay process is more probable per unit time. This often reflects a stronger effective coupling to the available final states, although the exact value also depends on kinematics and conservation laws.

So decay width is influenced by both the interaction responsible for the decay and the details of the decay products.

Final Summary

Decay width $\Gamma$ is a measure of how quickly an unstable particle decays. It is related to lifetime by

$$
\Gamma = \frac{\hbar}{\tau}
$$

It has units of energy. A larger width means a shorter lifetime. A smaller width means a longer lifetime. If several decay channels are possible, the total width is the sum of the partial widths,

$$
\Gamma = \sum_i \Gamma_i
$$

and each branching ratio is