violation. It is possible to derive evolution equations for the distribution functions in the variable $Q$ ($\log Q$ is often used instead) if one considers the various processes in the proton, such as quarks radiating gluons or gluons splitting into gluons or quark-antiquark pairs. Thus, if one knows the distribution function for a certain value of $Q$, the value for any other $Q$ can be calculated using these equations, which in the case of QCD are known as the *Altarelli-Parisi Equations*. These equations can be found in any book on Quantum Field Theory.

The distribution functions are important because the cross sections for different events (both in QED and QCD) can be expressed in these function. Thus, for the process we considered above, an electron hitting a proton, the cross section can be written as $$\sigma = \sum_{q,\bar{q}} \int dx[f_q(x,Q^2) \sigma (e^- q \to e^- q) +
f_{\bar{q}}(x,Q^2) \sigma (e^- \bar{q} \to e^- \bar{q})]
\label{eq:e1}$$ Here the cross sections in the integrand are the cross sections for the sub-collisions between the electron and the partons, and the sum runs over the quark and the antiquark flavors.

From now on we will use capital letters to denote the densities in the 