[MUSIC] In this fifth module we are discussing the structure of hadrons and strong interactions. In this second video we discuss inelastic scattering between electrons and nucleons and what one can learn from that. After following this video, you will know about resonances as excited states of nucleons. Dee inelastic scattering and nucleon structure functions. And the so.called scaling hypothesis as evidence for substructures inside the nucleon, as well as the role and distribution of quarks in the nucleon. Form factors decrease rapidly with q^2 like we showed in video 5.1. Consequently, the probability to observe an elastic scattering becomes low at high energy-momentum transfer. This is not surprising. Large q^2 corresponds to a short photon wavelength, which is then more and more able to resolve the internal structure of the target particle. This structure will then not be insensitive to the energy-momentum transfer. The target will be excited or even destroyed. In other words, inelastic processes take over. At the moderate q^2, inelastic processes produce excited states of the nucleon, also called resonances, such as âˆ†+, which have the same quantum numbers as the proton. These resonances an extremely short lifetime and therefore a broad mass distribution. The figure show the result of an electron-proton scattering experiment at an energy of about 3 GeV and at fixed scattering angle. The elasticity of the reaction is first visible in the energy distribution of the outgoing electron on the left, which is no longer fixed as in elastic case. Broader secondary maxima are formed which corresponded to the excited states of the nucleon, also clearly visible in the invariant mass distribution of the outgoing hadronic state on the right. Is kinematic is completely determined by measuring the outgoing electron as in the case of elastic scattering. The position and the inverse width of the large maximum indicate the mass and the life time of the resonance âˆ†+(1236). At higher q^2, beyond the resonance region we enter into the region of the so-called deep elastic scattering. The final state consists of multiple hadrons with at least one baryon to to conserve quantum numbers. This is the region where the photon interacts individually with the charged constituents of the nucleon. The cross section can be parametrized with 2 terms as before, but with 2 unknown structure function W_1 and W_2 replacing the form factors. They parametrize the distribution and dynamics of the nucleon constituents. They depend on the tranfered energy nu which is the energy of the photon and its invariant mass squared. The kinematics now requires a second additional kinematic parameter because the mass of the hadronic system is no longer fixed, as was the case with a proton or resonance in the final state. In addition to the scattering we can thus choose a second variable. Here it is the energy Eâ€™ of the outgoing electron. The structure functions are measured experimentally, analyzing the dependence of the cross section on the scattering angle, like in the elastic case. The big discovery at SLAC in the late 60s was that the structure functions do not depend on nu and q^2 separately but on their ratio x_Bj, which is equal to q^2 over twice the mass of the proton times nu. The variable is named after its inventor James D. Bjorken. The phenomenon was called scaling at that time because of the very general observation that the dimensionless variable x_Bj does not depend on any energy nor mass scale. Structure functions follow in fact scaling if inside the nucleon there exist sub-structures with which the photon interacts elastically. This interpretation of scaling is best demonstrated in the limit of large equal square, where the interaction between quarks is negligible compared to the force transmitted by the photon. Quarks then act as quasi free particles with which the photon interacts in an incoherent manner. To respect the kinematic constraint in such an elastic electron-quark scattering, the photon can only interact with a quark which carries a fraction x_Bj of the nucleon momentum. Consequently, the cross section is proportional to the probability to find such a quark in the nucleon. In the limit of square going to infinity, the structure functions W_i tend toward the function F_i(x) which are directly related to the distribution of the fractional momentum x_i carried by quarks of type i. The experiment at SLAC have in fact shown that the phenomenon already occurs at the moderate squared. The structure function W_2, as a function of the kinematical variable omega, which is about equal to 1/x_Bj, smoothens with increasing the q^2 beyond the resonance region, and then follows a universal function of x_Bj independent of q^2. The kinematic constraint imposed by elastic scattering of point-like particles inside the proton is indeed respected. However, the scaling of structure functions is not exact. This figure show modern measurements that come from experiments at the electron proton collider HERA. The curves and data for different x are expressed by a constant in order to be all visible in the same graph. At small x, so at small quark momenta, the structure function varies strongly with q^2 because of the contributions from virtual gluons enriching the quark contents. The smaller deviation at large x are more subtle. The coupling constant of strong interactions depends on the momentum transfer. It is not really a constant. We will come back to this astonishing fact in video 5.4. Structure functions can thus be used to measure the momentum distribution of quarks inside nucleons. We qualitatively demonstrate the behavior of the structure function F_2(x) based on successively refined hypotheses on the dynamics of quarks. If the nucleon consisted of three quarks without interaction, x will be fixed to 1/3 for each quark, and F_2 will be a delta function at this value. If it were a state where quarks are bound together via gluons, the average value of x would still be 1/3, but with a wider distribution around this value. If we finally take into account that quarks can emit virtual gluons which can split into a quark-antiquark pair at low energy, we expect that the quark distribution tends to fill up at x -> 0. The experimental data indeed to follow this qualitative picture. First of all for the valence quarks, that is the two up quarks and one down quark for the proton, the structure function varies around 1/3 with a strong enhancement at small x by additional quark-antiquark pairs produced by gluons. The integral of the distribution is twice as large for the up quark than for the down quark, as it should be. The contribution gluons of is indeed seen in the contribution of s and sbar quarks which would not be present in the nucleon otherwise. When integrating this distribution we see that only about 1/2 of the proton momentum is carried by quarks. The rest is carried by gluons. In the next video, we discuss mesons, bound states between quarks and antiquarks. [MUSIC]