  In recent years enormous progress has been made in perturbative quantum field
theory by applying methods of algebraic geometry to parametric Feynman
integrals for scalar theories. The transition to gauge theories is complicated
not only by the fact that their parametric integrand is much larger and more
involved. It is, moreover, only implicitly given as the result of certain
differential operators applied to the scalar integrand
$\exp(-\Phi_{\Gamma}/\Psi_{\Gamma})$, where $\Psi_{\Gamma}$ and $\Phi_{\Gamma}$
are the Kirchhoff and Symanzik polynomials of the Feynman graph $\Gamma$. In
the case of quantum electrodynamics we find that the full parametric integrand
inherits a rich combinatorial structure from $\Psi_{\Gamma}$ and
$\Phi_{\Gamma}$. In the end, it can be expressed explicitly as a sum over
products of new types of graph polynomials which have a combinatoric
interpretation via simple cycle subgraphs of $\Gamma$.
