New graph polynomials in parametric QED Feynman integrals

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 $\exp (-{\mathrm{\Phi }}_{\mathrm{\Gamma }}\mathrm{/}{\mathrm{\Psi }}_{\mathrm{\Gamma }})$, where ${\mathrm{\Psi }}_{\mathrm{\Gamma }}$ and ${\mathrm{\Phi }}_{\mathrm{\Gamma }}$ are the Kirchhoff and Symanzik polynomials of the Feynman graph $\mathrm{\Gamma }$. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from ${\mathrm{\Psi }}_{\mathrm{\Gamma }}$ and ${\mathrm{\Phi }}_{\mathrm{\Gamma }}$. 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 $\mathrm{\Gamma }$.