[MUSIC] Hello, and welcome to this fifth module of our introductory course to subatomic physics. In this module, we'll discuss the structure of hadrons and strong interactions. In this first video, we discussed the elastic electromagnetic scattering between electrons and nucleons, which is one of the ways used to study the structure of hadrons. After following this video, you will know the main properties of elastic scattering between fermions, and the meaning of the different terms that enter into the cross section. The form factor concept you will know, and its interpretation, in terms of the size of the nucleon. And you will get to know the historic experiments at the Stanford Linear Accelerator Center in this area. When the structure of an interaction is known, it can be used as a tool to better understand the particles that are involved. Within this video, how elastic electron nucleon scattering is used to determine the internal structure of nucleons. We take advantage of the fact that the electron as far as we know, is a point like particle in that we know electromagnetic interactions very well. We can thus elucidate a single, but important unknown, which is the distribution and the dynamics of quarks inside the proton and neutron, represented by the big blue dot in the right diagram. We’ll see in video 5.3, also, how meson resonances are formed by an incident photon, for example. These can be used to better understand the strong force that binds quarks and anti-quarks together. The prototype process for the scattering between point like fermions is electron muon scattering. It is very similar to e+ e- annihilation, with the Feynman diagram rotated by 90 degrees. We will approach this process in steps. The first step is already shown here, since we have encountered it in the past. It is to consider scattering of a very heavy type target, which is without structure and without spin. Also, ignoring the spin of the incoming electron, and neglecting its mass, we obtain the formula shown here, the Rutherford cross section, which we already encountered in video 1.4. As usual, the factor alpha squared comes from the two electromagnetic vertices. The denominator contains the square of the characteristic energy. It results from the propagator of the virtual photon, one over q^2, and you can easily verify this in the approximation that the mass of the electron is negligible, as I did here. The angular distribution is extremely steep for the scattering of point like particles, and it peaks at small angles. This remains valid when including the effect of the target recoil, and the spin of the particles. The additional term in red, shown here with M the mass of the target, takes into account the recoil. It is the ratio between the incident and the outgoing electron energy E’/E. This ratio is 1 for an infinitely heavy target, but less than 1 for an important recoil. This formula still neglects the magnetic component of the interaction. For a point like fermion target with a magnetic moment, µ = e/(2m). One finds the complete Mott formula given on the bottom of the page, quoted here in the laboratory frame, and for relativistic electron projectiles. The last term in red is of a magnetic nature. It becomes important at high momentum transfers q^2, much larger than the mass of the target squared The angular distribution, still roughly follows the form of the Rutherford formula, the third term contains sub-terms proportional to cosine squared theta, and to sine squared theta, which reduce a little the steepness of the angular distribution. The historic experiments on elastic scattering between electrons and protons have been conducted at the Stanford Linear Accelerator Center in the 1960s and 1970s. The experiment consisted of a liquid hydrogen target bombarded by electrons of a few hundred MeV. In the final state, only the electron is observed, in two spectrometers positioned at variable scattering angles. The kinematics of the hadronic final state is then deduced from energy-momentum conservation, assuming a target at rest. Here are some results on the cross section as a function of the scattering angle on the left, and as a function of the incident electron energy on the right. The measurements confirm the calculations at these modest energies, already below one GeV. One observes a steep angular distribution in the left plot, and a strong reduction of the cross section with energy in the right plot, as we expected from the formulae. Let us now consider what happens if the target is not a point-like particle. Let us first take the model of a static charge distribution rho(x), normalized such that the charge of the target remains elementary, which means that the integral over rho(x) must be equal to 1. The cross section then will be reduced with respect to a point-like target by a factor of F^2, a so-called form factor. We have already used this concept in video 2.2, when we discussed the size of nuclei. In the static case, F is simply the Fourier transform of the spatial charged distribution. For small momentum transfers, one can develop the form vectors in powers of q times x. If the distribution possess spherical symmetry, that is rho(x) is only a function of r, and not of the spatial directions separately, terms with odd exponent will not conribute. Thus, the form factor measures the mean square radius of the charge distribution, <r^2>, which represents the size of the distribution. Take, for example, an exponential distribution as a function of distance. It leads to the so-called dipolar form factor that is shown in the last formula on the right. It is clear that in the case of a non-static charge distribution, things get more complicated. First, magnetic interactions come in, since the target is moving. Second, the charge distribution itself changes during the time represented by the time-like component of the four-momentum transfer q_0. By the action of the moving charges inside the distribution, electric and magnetic terms will be modified, and their functions will in fact be mixed up. We then find the cross section given here with the parameter tau, which is the ratio between the momentum transfer and the target mass squared, the ratio of the two quantities, which characterize this process. We call G_E and G_M the electric and magnetic form factors, although the distinction between the two obviously depends on the reference frame. In the same sense, one can attribute G_E and G_M to the Fourier transforms of the charge and magnetic moment distribution of the target particle, even though this interpretation is, strictly speaking, only valid in a very special reference frame. The form factors G_E and G_M of the proton are measured by analyzing the differential cross sections for the reaction e- p -> e- p, and separating the terms proportional to cosine squared theta and sine squared theta of the angular distribution. We observe that G_E and G_M follow the dipolar shape, predicted for exponential charge distributions. The parameter Lambda then characterizes the size of the distribution, and we find Lambda equal to 0.84 GeV, in both cases. The size of the distribution is thus of order 1 fm and the same for electric and magnetic form factors. Finding the same distribution for both form factors is not a miracle. The quarks inside the nucleon are at the origin of both charge and magnetic moment distributions, such that we find the same distribution because they're caused by the same ingredient. The figure shows representative results obtained by Hoffstetter and collaborators at SLAC in the 1960s. The form factors G_E, shown on the left, and G_M, shown on the right, for the proton are measured as a function of the momentum transfer squared, and extracted from the cross section for the elastic electron proton cross section. One finds the dipolar shape, G equals one over one plus q^2 over Lambda squared, both squared, with the same size perimeter Lambda equal to 0.84 GeV. In the next video, we will see what happens when we increase q^2, such that the exchanged photon can excite, or even break the nucleon. [MUSIC]
