In a collider experiment, \(\Lambda\) baryons can be identified from the decay \(\Lambda \rightarrow \pi^{-} p\), which gives rise to a displaced vertex in a tracking detector. In a particular decay, the momenta of the \(\pi^{+}\)and \(p\) are measured to be \(0.75 \mathrm{GeV}\) and \(4.25 \mathrm{GeV}\) respectively, and the opening angle between the tracks is \(9^{\circ}\). The masses of the pion and proton are \(139.6 \mathrm{MeV}\) and \(938.3 \mathrm{MeV}\). (a) Calculate the mass of the \(\Lambda\) baryon. (b) On average, \(\Lambda\) baryons of this energy are observed to decay at a distance of \(0.35 \mathrm{~m}\) from the point of production. Calculate the lifetime of the \(\Lambda\).

In particle physics, the concept of invariant mass is crucial for understanding the properties of decaying particles. The invariant mass, also known as the rest mass, is a quantity that remains constant regardless of the reference frame. This is particularly important in high-energy physics, where particles move at velocities close to the speed of light.
Invariant mass can be computed using the energy \(E\) and the momentum \(p\) of the particles involved. For a system of particles, the invariant mass \(m\) is given by \[m = \sqrt{E^2 - p^2}\].
In the provided exercise, we calculate the mass of the \(\Lambda\) baryon using this formula. We first determine the total energy and momentum of the system from the detected particles (\(\pi^-\) and (\(p\)), then use these values to find the invariant mass.

Momentum is a fundamental concept in both classical and modern physics. It describes the quantity of motion an object has and is a vector quantity, meaning it has both magnitude and direction. In the context of particle physics, we deal with relativistic momentum, especially when particles are moving close to the speed of light.
For a particle with mass \(m\), moving with velocity \(v\), and in the relativistic regime, the momentum \(p\) is given by \[p = \gamma mv\], where \(\gamma\) is the Lorentz factor. The exercise makes use of the law of cosines to determine the total momentum of the decaying system. For particles with measured momenta and an opening angle between their tracks, the total momentum can be calculated accurately to help solve for other properties like invariant mass.

The Lorentz factor is a key concept in Special Relativity, describing how time, length, and relativistic mass change for objects moving close to the speed of light. It is defined as \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\], where \(v\) is the velocity of the object, and \(c\) is the speed of light.
In the given problem, after calculating the velocity of the \(\Lambda\) baryon, we use the Lorentz factor to determine the time dilation effects. This enables us to calculate the particle's lifetime from the observed decay distance. The Lorentz factor adjusts the time measured in the lab frame to the proper time of the particle's rest frame.

The lifetime of a particle is an important property that indicates how long it exists before decaying into other particles. This is often measured in terms of proper time, which is the time experienced by the particle in its rest frame.
To find the lifetime, we use a relationship that involves the observed decay distance and the velocity of the particle: \[\tau = \frac{d}{v} \cdot \frac{1}{\gamma}\], where \(d\) is the decay distance, \(v\) is the velocity, and \(\gamma\) is the Lorentz factor. By incorporating the relativistic effects via the Lorentz factor, we accurately account for how time dilates for fast-moving particles.
In the exercise, using the observed decay distance of the \(\Lambda\) baryon and its velocity, we compute its lifetime. This combined approach underscores the interplay between space, time, and high velocities in particle lifetimes.