Doubling bialgebras of graphs and Feynman rules

In this article, we define a doubling procedure for the bialgebra of specified Feynman graphs introduced in a previous paper \cite {DMB}. This is the vector space generated by the pairs $(\bar \Gamma, \bar \gamma)$ where $\bar \Gamma$ is a locally $1PI$ specified graph of a perturbation theory $\Cal T$ with $\bar \gamma \subset \bar \Gamma$ locally $1PI$ and where $\bar \Gamma / \bar \gamma $ is a specified graph of $\Cal T$. We also define a convolution product on the characters of this new bialgebra with values in an endomorphism algebra, equipped with a commutative product compatible with the composition. We then express in this framework the renormalization as formulated by A. Smirnov \cite [\S 8.5, 8.6] {Sm}, adapting the approach of A. Connes and D. Kreimer for two renormalization schemes: the minimal renormalization scheme and the Taylor expansion scheme. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure, by following the approach by P. Etingof \cite{PE} (see also \cite{RM}).