  We study renormalization in a kinetic scheme using the Hopf algebraic
framework, first summarizing and recovering known results in this setting. Then
we give a direct combinatorial description of renormalized amplitudes in terms
of Mellin transform coefficients, featuring the universal property of rooted
trees H_R. In particular, a special class of automorphisms of H_R emerges from
the action of changing Mellin transforms on the Hochschild cohomology of
perturbation series.
  Furthermore, we show how the Hopf algebra of polynomials carries a refined
renormalization group property, implying its coarser form on the level of
correlation functions. Application to scalar quantum field theory reveals the
scaling behaviour of individual Feynman graphs.
