unique algebra homomorphisms $
\phi_-: \calh\to \CC+P(R)$ and $\phi_+: \calh\to \CC+(\id-P)(R)$ such that $$\phi=\phi_-^{\star (-1)}\star \phi_+.
\mlabel{eq:abd}$$ Here $\star$ is the convolution product and $\phi_+$ is called the [**renormalization**]{} of $\phi$. This algebraic setup is reviewed in Section  together with a discussion of quasi-shuffle algebras.

To apply this setup to the renormalization in QFT, one takes

-   $\calh=\calh_\FG$ to be the Connes-Kreimer Hopf algebra of Feynman diagrams, parameterizing regularized Feynman integrals,

-   $R=\CC[[\vep,\vep^{-1}]$ to be the Rota-Baxter algebra of Laurent series, and

-   $\phi$ to be the the regularized Feynman rule that assigns a Feynman diagram to the Laurent series expansion of the corresponding regularized Feynman integral.

Then the renormalized values of a Feynman integral is given by $\phi_+(\Gamma)$, where $\Gamma$ is the corresponding Feynman graph, when $\vep$ approaches zero. For further details see .

To apply this setup to our study of renormalized MZVs, we similarly define

-   $\calh$ to be the quasi-shuffle Hopf algebra parameterizing regularized MZVs,

-   $R$ to be the Rota-Baxter algebra $\CC[T][[\vep,\vep^{-1}]$ of log Laurent series, and

-   $\phi$ to be the algebra homomorphism sending a symbol in $\calh$ to the Laurent series expansion of the corresponding regularized MZV.

Once these are obtained in Section 