Five pages  Page one University of the Witwatersrand Advanced Techniques in Physics : 2003 Examination : June 2003 
The first few years of LEP running were devoted to the study of production and decay with the energy of the accelerator being varied in small energy steps around the resonance ( invariant mass).
One of the important results has been the determination of a the number of lepton families to a certain confidence limit. Figure illustrates this point. One can see that only the theory for three families of particles correctly describes the data.
In this project, you will have to verify the conclusion that only three families of particles describe the data, and in addition, determine the confidence limits of this statement. You will also have to consider the assumptions underlying this conclusion, so that you can comment on how reliable it can be expected to be.
The absolute crosssection (probability) for the processes
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In the three family model, is either a lepton: , or , a neutrino: , or , or one of the five quark flavours: , , , or . (Only fermions with masses contribute directly to the decay crosssection.)
The theoretical calculation of the crosssection at the lowest order is done using the first Born approximation. At this level the resonance crosssection is a BreitWigner Lorentzian. Without the radiative corrections, however, the predicted line shape is grossly inadequate. The radiative corrections modify the predictions for the partial widths, and introduce an energy dependence in the resonance width. The detailed shape of the resonance is therefore sensitive to many aspects of the Standard Model. This includes constraining the top mass, QCD checks, measurements of (the fine structure constant), setting lower limits on the Higgs mass and so on. Out of this wealth of information available in this data, for our purposes we are only interested in measuring, with confidence limits, the number of particle families.
Considering the above, the new form for the resonance crosssection
becomes :
A significant correction (up to 30%) at the resonance energy, is left out of this expression. The electronpositron pair in the entrance channel bremsstrahl photons which lower the centre of mass energy available for production. This causes a high energy tail as well as a lowering of the peak in the resonance crosssection. Consider the raw data of figure and convince yourself that you see this effect. The BreitWigner Lorentzian line shape given above would therefore not be adequate, and would require modification in order to correctly model the observed line shape. These modifications have been computed, and are available in the literature.
From the LEP results it can be well established that there are no charged leptons from possible additional families with masses , since these would have been seen.
There could conceivably be many families (quarks and charged leptons). However, due to the condition , only the neutrinos of these other families are usually expected to contribute to the total width, . The reasoning, in the case of a possible fourth particle family, is that the lepton and quarks for this family may be massive enough not to appear in the calculation of the line shape. However, the neutrino of this possible fourth family is unlikely to be massive enough to be excluded (neutrinos are at least rather light).
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is the total hadronic width, is the width due to each of the charged lepton channels and is the width for each neutrino channel. If there are more than three families, then
Electroweak predictions for the relative rates of the channels are as follows:
each neutrino channel  6.7% 
each charged lepton channel  3.35% 
total hadronic channels  70% 
From this it can be seen that each additional neutrino channel will increase by 6.7%. The decay events can be observed and sorted according to the exit channel represented by separately, except for the neutrino exit channel. Hadronic data will of course have the better statistics. Hadronic data (one selects a subset of the observed events where the primary particles in the exit channel are hadrons) is therefore used to provided the most stringent evaluation of the number of fermion families. Hadronic data for production is reproduced in the table below. It has however been doctored. The effect of entrance channel bremsstrahlung (discussed above) has been unfolded from the raw data according to a model for it, so that the simple form of presented above (which is not corrected for entrance channel bremsstrahlung) can nevertheless still be used. This has been done by the examiner to simplify the question. In practise, one prefers not to doctor data, and rather use a more complex theory in the description of the line shape.

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