Phenomenology

The superposition of transparencies for the audiovisual impact cannot be replaced but it cannot either be reproduced in a text like this. Nevertheless, it is necessary to indicate, even without much didactic success, two or three experimental evidences where the fundamental interactions such as we understand them today are the protagonists.

- We will begin by the forward-backward asymmetry in the reactions $e^{+} e^{-} \rightarrow f \bar{f}$ where, for example $f=\mu^{-}$. The purely electromagnetic differential cross section corresponding to the diagram of Figure 2 is

Figure 2: Electron-positron annihilation in a pair of fermions and its kinematics

$$\frac{d\sigma}{d\Omega} = \frac{e^{4}}{64 \pi^{2} s} (1 + \cos^{2}\theta)$$ (1)

The experimental data indicate that another contribution is present. It is the weak interference due to the diagram in Figure 3

Figure 3: Weak contribution to electron-positron annihilation in a pair of fermions

that corresponds to the neutral weak boson $Z$ which, due to its characteristic coupling gives place to

$$\frac{d\sigma}{d\Omega} = f(\cos^{2}\theta, \cos\theta)$$ (2)

providing a forward-backward asymmetry through the $\cos \theta$ dependence, in very good agreement with experimental data as Figure 4 shows.

Figure 4: Annihilation angular distribution. QED and electroweak prediction

- Another important evidence emerges from the so called $R$ relation between the cross section for electron-positron annihilation giving hadrons and the corresponding to $\mu^{+} \mu^{-}$ , that is to say

$$R = \frac{\sigma (e^{+} e^{-} \rightarrow hadrons)}{\sigma (e^{+} e^{-} \rightarrow \mu^{+} \mu^{-})}$$ (3)

This relation within the context of the parton model, can be represented by the diagram in Figure 5 which evidently takes us to the value:

Figure 5: Quark-parton model contribution to $R$

$$R_{QPM} = 3 \sum Q_{f}^{2}$$ (4)

where $3$ is for the colour degree of freedom and indicates at this level the crucial presence of a feature (at least ``kinematic") of the strong interactions in the middle of a situation a priori electromagnetic. Nevertheless, the data show that this strong presence should be even bigger. A more detailed calculation gives for the corresponding corrections related to the diagrams in Figure 6

Figure 6: First order QCD corrections

the expression:

$$R_{QCD} = R_{QPM} \{1 + \frac{\alpha_{s}}{\pi} + C_{2} \fra... ...^{2} + \cdots \} \simeq R_{QPM}\{1 + 0.05 + 0.004 + \cdots \}$$ (5)

with

$$C_{2} (\overline{MS}) = 1.986 - 0.115 N_{f} \;;\; \hspace{0.2cm}(N_{f}: number of flavors)$$ (6)

In spite of these corrections, the agreement with the experimental data is not adequate unless the weak corrections coming from the diagram of Figure 3 are entered. They lead to the prediction

$$R = 3 \sum Q_{f}^{2} \{1 + C_{1}^{V} (\frac{\alpha_{s}}{\pi}... ...V} \frac{\alpha_{s}}{\pi}^{2}) C_{VV} + [ V \rightarrow A ]\}$$ (7)

with

$$C_{VV} = Q_{f}^{2} - 2 Q_{f} v_{e} v_{f} Re(\chi) + (v_{e}^{2} + a_{e}^{2}) v_{f}^{2} \mid \chi\mid^{2}$$ (8)

$$C_{AA} = (v_{e}^{2} + a_{e}^{2}) a_{F}^{2} \mid \chi \mid^{2}$$ (9)

$$\chi(s) = \frac{G_{F}}{8 \sqrt{2} \pi \alpha}\; \frac{s M_{Z}^{2}}{s - M_{Z}^{2} + i M_{Z} \Gamma_{Z}}$$ (10)

that now does reproduce the experiment as shown in Figure 7

Figure 7: R as a function of energy

- Let us go now to the jets, stream of particles produced in different types of reactions. For example the one in Figure 8

Figure 8: Hadronic jets in electron-positron annihilation

The quarks of the final state are materialized in hadrons which arise in bunches around the direction defined by either $q$ or $\bar{q}$. The Figure 9

Figure 9: Two jets event

reproduces the image of one of these events in a detector (Jade in this case), where the grouping of the hadrons in the final state is evident. Here is very clear the presence at short distances (the initial instant) of the quarks defining directions.

Figure 10: Origin of three jets

If a quark emits a (sufficiently hard) gluon before hadronization, a third direction can be defined as in Figure 10

Figure 11: Three jets event

In this case, events will be produced, as the one represented in the Figure 11 (from Tasso detector) with 3 jets: the presence of the gluon now appears evident.

All the evidences that we have selected have the objective of inducing phenomenological reasons to formalize a theoretical model. This model should not only contain this information but also have a predictive value, being at the same time coherent from the formal point of view. The introduction of such model is the next goal. The mentioned evidences were also thought to provoke amazement and so predispose ourselves for science because: ``It is precisely a characteristic of the philosopher this state of mind: the amazement, which is the principle of philosophy..."(Plato, Teeteto, 115d)