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I'm currently trying to understand and reproduce the Hopf-algebraic renormalization of QED presented by Walter D. Van Suijlekom. I don't understand why he chooses these external structures in (2), (7), (8), (10): $$ \langle\sigma_{(1)},f\rangle = \frac{1}{16}\mathrm{Tr}[\gamma_\mu f(q=0,\mu;p=0)]\\ \langle\sigma_{(0)},f\rangle = \frac{1}{16}\mathrm{Tr}[\gamma_\mu \frac{\partial}{\partial p_\mu} f(p)]|_{p=0}+m^{-1}f(p=0)\\ \langle\sigma_{(2)},f\rangle = \frac{1}{16}\mathrm{Tr}[\gamma_\mu \frac{\partial}{\partial p_\mu} f(p)]|_{p=0}\\ \langle\sigma_{(3)},(-\delta^{\mu \nu}q^2+q^\mu q^\nu)\otimes M\rangle = \mathrm{Tr}[M]\\ $$

Note that we work on the Euclidean configuration, in Minkowski we have to consider $\delta_{\mu \nu} -> g_{\mu\nu}$.

Starting by the $\sigma_{(1)}$ case, $f$ will be replaced by a truncated vertex diagram amplitude later on (at the renormalization time of some diagram $\Gamma$). For $\sigma_{(0)}$, $\sigma_{(2)}$, $f$ will be replaced by a truncated self-energy amplitude. For $\sigma_{(3)}$, $(-\delta^{\mu \nu}q^2+q^\mu q^\nu)\otimes M$ will correspond to a truncated vacuum polarization amplitude.

All of these values will correspond later on to multiplicative factors in the counterterms of the BPHZ algorithm. These must have a direct equivalent in BPHZ renormalization of QED, but I am not being able to find it anywhere. Thank you so much.

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