[MUSIC] Hello, and welcome to this fourth module of our introductory course on subatomic physics. This module starts a series of three modules, where we talk about the three fundamental forces described by the standard model of particles physics. During this first module, we go into more details about the electromagnetic interactions and their properties. You will notice that the intellectual challenge, and also the mathematical levels is slightly higher than in previews modules. But in this first video, we will remind you how to describe the intensity of the reaction using the cross section and the decay rate, so that you are up to speed with the rest of the course. After following this video, you will know the relation between the reaction rate and the cross section. Also, the relation between the decay rate, and the lifetime of a particle. And how the probability amplitude for a process enters into all these notions. So, we now resume our discussion of scattering and decay processes. To characterize these processes, we need a quantity which measures the intensity of a reaction or, in the context of quantum mechanics, its probability. This quantity is the cross section sigma that we introduced in video 1.3 for scattering processes, and it's analog for decays, which is called the decay rate. Both quantities measure the ratio of the number of scatters or decays, that actually take place, to the number that would have been possible, that could have taken place. If this were a game, we would speak of this probability as the ratio between the successes and the trials of the game. We have already seen that the unit for the cross section is a bit surprising, it is a surface. We will see that for the decay rate, it is an energy. This is due to using the system of natural units that we have introduced in video 1.2a. If you are still somewhat uneasy with the construction an interpretation of Feynman diagrams, like the ones that you see behind me, – on the left for a scattering process on the right for a decay process – you should follow the optional video 4.1a. As quantum physics takes a statistical approach, the basic problem of particle physics is the measurement and calculation of the probability for a process. Both calculation and measurement use the cross section to express this probability independently of experimental details. We consider here a four-body reaction a + b -> c + d. If each target particle b has a cross section sigma, which represents the probability to hit it, the rate of scatters per second will be proportional to the incident flux of projectiles and the number of targets. The cross action is thus defined as the reaction rate per target particle, normalized to the flux of projectiles. It represents the intensity of a reaction independent of exponential parameters. The proportionality factor between the cross section and the rate of interaction is called luminosity. The laboratory frame, the flux of project ions I_a, that is to say their number per unit surface and per unit time is equal to their volume density times their velocity relative to the targets b, which are at rest. The luminosity L is the product of this flux, and the number of targets N_b. In a colliding beam experiment on the country, the luminosity is the product of the number of projectiles and targets per bunch, N_a and N_b, multiplied by the crossing frequency f and divided by the surface, that is common to the two beams. The unit of the cross section is the barn. In high energy reactions, we rather find cross section of the order of the nanobarn, or even picobarn. The smallest known cross sections are those of neutrinos, of the order of attobarn. As far as decays are concerned, the description is quite similar. There's of course no projectiles, so we only consider the particles b in the initial state. We normalize the number of decays per second to their current number N_b, to find the decay rate Gamma. It is the inverse of the lifetime tau_b. The decay rate Gamma has the dimension of an energy. It is also called the width of the particle, because it corresponds to the uncertainty in the mass of b particles. Therefore, it can be measured either by measuring the lifetime of the particle, or by observing the width of the invariant mass distribution of its decay products. As the decay rate is a constant, we find an exponential law for the number of remaining particles at time t. Consequentially, also, the absolute number of decay in time interval ∆t decreases exponentially. If several decay channels exist, the lifetime observed in each channel is of course the same, but the decay rate depends on the channel. One thus defines a partial decay rate Gamma_i, for each channel i. By probability conservation the sum is equal to the total width Gamma. The branching ratio Gamma_i/Gamma expresses the relative probability of observing channel i, among all decay channels. Analogous to the differential cross section, one can also measure the differential decay rate, dGamma/dOmega, depending on the solid angle. Let us think for a moment. If, for a decay with multiple decay channels, we only observe one channel i. Is the time evolution determined by Gamma, or is it determined by Gamma_i? Well, it is, of course, determined by Gamma. Since the lifetime of the particle is independent of the decay channel, that it will choose. In the next video, we will apply the concept of cross section to elementary electromagnetic interactions. [MUSIC]
