  The main theme of the paper is the detailed discussion of the renormalization
of the quantum field theory comprising two interacting scalar fields. The
potential of the model is the fourth-order homogeneous polynomial of the
fields, symmetric with respect to the transformation
$\phi_{i}\rightarrow{-\phi_{i}}$. We determine the Feynman rules for the model
and then we present a detailed discussion of the renormalization of the theory
at one loop. Next, we derive the one loop renormalization group equations for
the running masses and coupling constants. At the level of two loops, we use
the FeynArts package of Mathematica to generate the two loops Feynman diagrams
and calculate in detail the setting sun diagram.
