Self-Similarity and Criticality in the Boson (pions) field produced according to dynamic cascade models

In this work we reveal that the inside-outside dynamic model for the production of fermion-antifermion pair (or quark-antiquark in QCD), is an out of equilibrium process. Properties of a complex system as the self-similarity and the critical point of a continuous phase transition in equilibrium appear. In this work we show that this is accomplished imposing a type of mapping process through a self-composition operator on a QED_2 bosons (or pions) field.