beta functions in the quantum field theory

Introduction/purpose: the running of the coupling constant in various quantum field theories and a possible behaviour of the beta function are illustrated. methods: the callan-symanzik equation is used for the study of the beta function evolution. results: different behaviours of the coupling constant for high energies are observed for different theories. the phenomenon of asymptotic freedom is of particular interest. conclusions: quantum electrodynamics (qed) and quantum chromodinamics (qcd) coupling constants have completely different behaviours in the regime of high energies. while the first one diverges for finite energies, the latter one tends to zero as energy increases. this qcd phenomenon is called asymptotic freedom.