Particle Lifetimes and Decay Rates
A particle's lifetime ( \( \tau \) ) is the average amount of time it exists before decaying. It provides a statistical measure of the particle's stability and is inversely proportional to its decay rate. $$ \tau = \frac{1}{\Gamma} $$
When an unstable particle is created, there is no way to predict exactly when it will decay. Particle decay is fundamentally a probabilistic process.
For example, two muons produced at the same instant can decay at completely different times.
Because of this randomness, physicists do not focus on the lifetime of a single particle. Instead, they study the average lifetime observed across a large number of identical particles.
The lifetime of a particle is conventionally denoted by the symbol \( \tau \).
Understanding particle lifetimes is one of the key goals of particle physics. It allows us to describe how unstable particles transform into other particles and how likely different decay processes are to occur.
Note. Particle physics mainly deals with three broad categories of physical systems and processes: bound states, decays, and scattering processes. Non-relativistic quantum mechanics provides an effective description of many bound states, while relativistic quantum field theory is the fundamental framework used to describe particle decays and high-energy scattering phenomena.
Since decay is a probabilistic phenomenon, the outcome of an individual decay event cannot be predicted in advance. However, by studying large numbers of particles, physicists can determine a particle's lifetime from its experimentally measured decay rate.
The decay rate, denoted by \( \Gamma \), represents the probability per unit time that a particle will decay. In quantum field theory, this quantity is also known as the decay width.
An important feature of particle decay is that the probability of decay does not depend on the particle's age.
A particle that has just been created and one that has existed for a much longer time have exactly the same probability of decaying during the next instant. In other words, particles have no memory of how long they have existed.
For example, a newly created muon and an older muon are equally likely to decay in the next moment.
Note. This behavior is very different from what we observe in everyday life. An 80-year-old person generally has a shorter life expectancy than a 20-year-old person. Unstable particles do not behave this way. For a particle, age does not matter. A "young" particle and an "old" particle always have the same probability of decaying during a given interval of time.
As a result, a particle's lifetime is inversely proportional to its decay rate.
$$ \tau = \frac{1}{\Gamma} $$
A large decay rate corresponds to a short lifetime, while a small decay rate corresponds to a long lifetime.
For example, suppose the decay rate of a muon is
\[ \Gamma = 0.1 \ \text{s}^{-1} \]
The corresponding lifetime is
\[ \tau=\frac{1}{\Gamma}=\frac{1}{0.1}=10 \ \text{s} \]
This does not mean that every muon survives for exactly 10 seconds. Instead, it means that the average lifetime measured across a large population of muons is 10 seconds.
Note. The numbers used in this example have been chosen to keep the calculation simple. In reality, the lifetime of a muon is much shorter, approximately \( 2.2 \times 10^{-6} \) seconds, or 2.2 microseconds. Even so, saying that a muon has a lifetime of \( 2.2 \times 10^{-6} \) seconds does not mean that every muon survives for exactly that amount of time. Decay is a random process. One muon may decay after \( 10^{-7} \) seconds, another after \( 5 \times 10^{-6} \) seconds, and another after \( 10^{-5} \) seconds. What can be predicted is not the behavior of an individual particle, but the statistical behavior of a very large population of particles.
The number of surviving particles decreases exponentially with time according to the exponential decay law.
The decay equation is
\[ N(t)=N(0)e^{-\Gamma t} \]
where \( N(0) \) is the initial number of particles, \( N(t) \) is the number of particles remaining at time \( t \), and \( \Gamma \) is the decay rate.
This equation shows that the number of surviving particles falls exponentially as time passes.
For example, suppose we start with \( N(0)=1000 \) particles and a decay rate of \( \Gamma = 0.1 \ \text{s}^{-1} \). After \( t=10 \) seconds, approximately 368 particles remain.
\[N(t)=N(0)e^{-\Gamma t} \]
\[ N(10)=1000 \cdot e^{-0.1 \cdot 10} \]
\[ N(10)=1000 \cdot e^{-1} \]
Since \( e^{-1}\approx 0.367879 \), it follows that
\[ N(10)\approx 1000 \cdot 0.367879 \]
\[ N(10)\approx 367.879 \]
\[ N(10)\approx 368 \]
Therefore, after 10 seconds approximately 368 particles remain, while about 632 particles have decayed.
Derivation. Suppose that \( N \) particles are present at time \( t \). During an infinitesimal time interval \( dt \), the number of particles that decay is proportional both to the number of particles present and to the decay rate \( \Gamma \). The fundamental decay law is \[ dN=-\Gamma N\,dt \] The negative sign indicates that the number of particles decreases over time. Solving this differential equation leads directly to the exponential decay law \[ N(t)=N(0)e^{-\Gamma t} \]
Many particles can decay in more than one way. These different possibilities are known as decay channels.
Each decay channel is characterized by its own partial decay rate \( \Gamma_i \).
For example, the positively charged pion decays predominantly through the channel \( \pi^+ \rightarrow \mu^+ + \nu_\mu \), although several less probable decay modes are also possible.
If a particle can decay through \( n \) different channels, the total decay rate is the sum of the partial decay rates associated with all channels.
\[ \Gamma_{\text{tot}}=\sum_{i=1}^{n}\Gamma_i \]
The particle's lifetime therefore depends on the total decay rate:
\[ \tau=\frac{1}{\Gamma_{\text{tot}}} \]
Consequently, determining a particle's lifetime requires taking all possible decay channels into account.
Example. Suppose a particle can decay through two channels:
- Channel A: \( \Gamma_1 = 2\ \text{s}^{-1} \)
- Channel B: \( \Gamma_2 = 3\ \text{s}^{-1} \)
The total decay rate is
\[ \Gamma_{\text{tot}}=\Gamma_1+\Gamma_2=(2+3)\,\text{s}^{-1}=5\ \text{s}^{-1} \]
The corresponding lifetime is
\[ \tau=\frac{1}{\Gamma_{\text{tot}}}=\frac{1}{5}=0.2\ \text{s} \]
Although each channel contributes differently to the decay process, the lifetime depends only on the total decay rate obtained by summing the contributions from all available channels.
Branching Ratios
In addition to the lifetime, it is often important to know how likely each decay channel is relative to the others.
This quantity is known as the branching ratio.
For the \( i \)-th decay channel, the branching ratio is defined as
\[ BR_i=\frac{\Gamma_i}{\Gamma_{\text{tot}}} \]
The branching ratio represents the fraction of all decays that occur through a particular channel.
A Practical Example
Suppose a particle has two decay channels:
\[ \Gamma_1=6 \]
\[ \Gamma_2=4 \]
The total decay rate is
\[ \Gamma_{\text{tot}}=6+4=10 \]
The branching ratios are
\[ BR_1=\frac{6}{10}=0.6 \]
\[ BR_2=\frac{4}{10}=0.4 \]
This means that 60% of all decays occur through the first channel, while 40% occur through the second channel.
Branching ratios therefore tell us how frequently a particle decays through each available channel.
Conclusion
In practice, two quantities are sufficient to describe most aspects of particle decay:
- Total decay rate and lifetime
How quickly the particle decays (\( \Gamma \) or \( \tau \)) - Branching ratios
Which final states are produced and with what probability
The lifetime is determined by the total decay rate, while the branching ratios describe how the decay is distributed among the various possible channels.
Together, these quantities provide a complete statistical description of how unstable particles decay.
