[MUSIC] In this fourth module, we are in the process of discussing the properties of electromagnetic interactions. And in this fifth video, we discuss the annihilation of electron-positron pairs into pairs of fermions. After following this video, you will know the main properties of the process of e+ e- annihilation into lepton pairs, and also the peculiarities of their annihilation into hadrons. The annihilation of electron-positron pairs into pairs of photons that Mercedes has introduced int the previous video is not the only annihilation reaction. There's a whole class of reactions of electron-positron annihilation into fermion-antifermion pairs and the example is shown here. We do not include the reaction e+ e- -> e+ e-, which is complicated by a second scattering diagram where the two electrons exchange a photon rather than annihilate. Let us take the creation of pairs of muons as a prototype. In the center-of-mass frame the final state is characterized by two muons with equal and opposite momentum. Having no strong interactions and being too heavy to cause electromagnetic showers, muons penetrate all the material in the detector, just losing energy by dE/dx. The total cross section again exhibits the well known factors, alpha squared in the nominator and s, the square of the characteristic energy, in the denominator. These are the characteristic factors for electromagnetic reactions between point-like particles. The cross section for tau production, e+ e- -> tau+ tau-, is the same, far from threshold, that is to say, at s, much larger than m_tau^2. The differential cross section is symmetric with respect to the sign of the scattering angle theta, if one neglects the masses, s >> 4 m_µ^2. The figure shows the angular distribution at high energy, the dashed lines indicate the shape of the symmetric distribution predicted by quantum electrodynamics. The solid lines include an asymmetric term that comes from electro-weak interference. The angular distribution in fact shows a deviation from symmetry that increases with energy. It is due to the interference with weak interactions that we will treat in module 6. This prototype cross-section applies to all annihilation reactions which produce a fermion pair of negligible mass and charge ±e. The coupling constant is hidden in the fine structure constant, alpha = e^2/(4π). Again, an identical angular distribution is observed for the reaction e+ e -> tau+ tau- at energies much larger than the threshold for this reaction. Let us now turn to pair production of quarks, or rather quark-antiquark pairs by e+ e- annihilation. Here we must consider two important changes. First, quarks carry electric charge, either +2/3 for u, c and t quarks or -1/3 for d, s, and b quarks, relative to the elementary charge e. As the quark charge comes in at only one of the two vertices, we must introduce a factor Q_i^2 in the cross section for each quark type i. Second, quarks are produced with three color charges, red, green or blue, charges that are responsible for the strong interactions but to which electromagnetic interactions are insensitive. The color of quarks is another example of the property of the final state which is observable only in principle. We have no way to measure it, but it distinguishes nevertheless final states corresponding to different colors. We must therefore add the amplitudes squared or the cross-sections. These are obviously independent of color, and for each flavor we obtain a cross-section which depends only on the electric charge, but is three times larger than if the object had no color. The mass of quarks is negligible for u, d, and s, but not for the heavy quarks c, b, and especially t. For them, their production thresholds, as indicated on the bottom of the slide, in the inclusive cross section e+ e- -> q qbar, there are steps at these thresholds. Quarks in the final state are not observed as such, because of their strong interaction which starts acting as soon as they are produced. The color field that is established between them is so energetic that additional quark- antiquark pairs spontaneously pop-up and cluster around the primordial quark as hadronic jets. You see an example in this picture from the L3 experiment at LEP. The jets follow the initial direction of the quark. Their total energy is that of the quarks. The multiplicity of charged and neutral particles inside each jet is high. Each quark forms at least one jet such that the event contains at least two of them. We will come back to the jet phenomenon in module 5 when we discuss strong interactions. Because of the conversion of quarks into hadrons, we cannot necessarily differentiate their flavor either. We consider therefore often the inclusive hadron cross section sigma(e+ e- -> hadrons), which to first order is given by the sum of the individual cross sections. This again because in principle, flavor allows to distinguish them. You may wonder if the conversion of quarks into hadron jets does not change the cross-section. This is not the case because as far as we know, this conversion is always happening, it has probability one. In other words, the quarks are always, and without residues, converted into hadrons. The cross section thus does not change. To compensate for the important reduction with the square of the energy, proportional to 1/s, of the electron positron cross-section, we often form the ratio R of the hadronic cross-section to that of our reference process, e+ e- -> µ+ µ-. The sum includes all flavors, as indicated in this equation, that can be produced at the given energy in the center of mass frame. To first order in the different energy regions we just obtain simple ratios of whole numbers. At each threshold there is a step in the cross section because a new channel opens up. This figure compares the rough calculation that we just did to a compilation of experimental results from the particle data group. The data comes from experiments at various e+ e- colliders, at different energies. We see the gross features of the cross section as predicted, including a small step at the quark threshold. But the measured cross section are significantly larger than the simple fractions we have just calculated. This is again due to higher order contributions. We will see in the next module that final states with additional gluons add some 10% to the total inclusive hadron cross-section. At high energies, above few tens of GeV, we see the influence of the weak neutral interaction. The exchange of its intermediate boson, the Z boson, dominates the total cross section at these energies, and causes the big peak that you see on the right of this graph. We will see much more of this in module 6. But before that, in the next module we'll discuss strong interactions and hadronic structure. [MUSIC]