The existence of the antihyperons was proven by the...

The existence of the antihyperons was proven by the discovery of an antilambda by M. Baldo-Ceolin and D. J. Prowse in 1958. A beam of negative pions with energy $E_\pi=4.6 \mathrm{GeV}$ hit an emulsion stack. What is the final state containing a $\bar{A}$ that can be produced in a $\pi^{-} p$ collision at minimum energy? Find the threshold energy if the target protons are free and, approximately, if they are bound inside nuclei with a Fermi momentum $p_{\mathrm{f}}=150 \mathrm{MeV}$. Assuming that the pion beam was produced at a distance $l=8 \mathrm{~m}$ upstream of the emulsion and that the number of produced pions was $N_0=10^6 / \mathrm{cm}^2$, how many pions $/ \mathrm{cm}^2$ reached the emulsion?

Key Concepts

Beam Attenuation and Particle Survival

As a beam of particles travels from the production point to the target, processes such as decay and interactions with the surrounding medium reduce the number of particles that actually reach the target. This concept involves calculating the reduction in beam intensity based on the distance traveled and the lifetime (or decay probability) of the particles, which is crucial for determining the effective number of particles arriving at the reaction site.

Fermi Motion in Nuclear Targets

In a nucleus, protons and neutrons are not at rest but move with momenta up to the Fermi momentum. This intrinsic motion (Fermi motion) alters the energy balance in collision reactions. When the target nucleon is bound inside a nucleus, its non-zero momentum can effectively lower the additional kinetic energy required from the incoming particle to reach the threshold for reaction, compared to a free proton target.

Conservation Laws in Particle Physics

Energy, momentum, and quantum numbers (such as baryon number and charge) must be conserved in any interaction or reaction. When producing an antimatter baryon, the reaction must also create an accompanying baryon to conserve baryon number, and the net charge must balance. These conservation laws are fundamental in establishing the allowed final states and calculating the minimum energies needed.

Invariant Mass and Center?of?Mass Frame

The invariant mass is a quantity calculated from the energy and momentum of a system that remains the same in all inertial frames. Determining the invariant mass in the center?of?mass frame is crucial for computing the threshold energy because it sets the minimum total energy required for a particular final state. This approach simplifies the analysis of reaction kinematics, ensuring that energy and momentum are conserved in the collision.

Reaction Threshold Energy

This concept refers to the minimum energy that must be available in the laboratory frame for a specific reaction to occur. In particle collisions, especially when producing new particles, energy and momentum conservation dictate that the incoming particle must have sufficient energy to create the rest mass of the produced particles. For pair production, the threshold is reached when the center?of?mass energy is just enough to produce the two particles at rest.

Transcript

00:01 We are interested in this reaction in the laboratory.

00:05 They give us masses, these are approximate, we know the proton is 938 for instance.

00:19 We want to figure out the threshold.

00:21 We are supposed to do that by working in the center of mass frame.

00:27 So in the center of mass frame at threshold the k and the sigma particles are released at rest.

00:39 So we know that after the collision the total energy will just be the sum of their rest energies.

00:51 And we also know after the collision, excuse me, before the collision that the magnitude of the momentum of the proton and the pion are going to be equal and also they are going to be basically just moving in the x direction.

01:10 There are no angles to worry about.

01:12 Again because we are just looking for a threshold.

01:19 So that is what our formula looks like.

01:22 It should be the absolute value of the velocities because they are in opposite directions, those two velocities.

01:32 And at the threshold our total energy which is the energy of the proton plus the energy of the pion in the center of mass frame.

01:46 So these gamma factors are the gamma factors of the proton and pion in the center of mass frame.

02:01 And we can figure out, we can get, write gamma beta in terms of just gamma.

02:15 So we get this equation for gamma p and gamma pi.

02:25 So we have a pair of equations and two unknowns for those two.

02:28 They are not linear though...

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