New graph polynomials in parametric QED Feynman integrals

Abstract

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(−, where Ψ Γ and Φ Γ are the Kirchhoff and Symanzik polynomials of the Feynman graph Γ. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from Ψ Γ and Φ Γ . 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 Γ.

Dodgson polynomial identities

“…where X D is a product of the polynomials from [20] andc(D) is an integer determined by the combinatorial properties of D. In our main results, theorems 3.1 and 3.8, we prove that the sums in I (0) Γ and I (1) Γ are equal to a simpler sum of the form…”

“…). They also satisfy the contraction-deletion relations and the following three useful identities (proposition 2.8 and lemmata 2.9, 2.10, 2.11 in [20]):…”

“…The sets of bonds and simple cycles of a graph G are denoted B G and C [1] G . In [20] two polynomials based on these types of subgraphs were defined and it was shown that they can be used to express the Schwinger parametric integrand in quantum electrodynamics without derivatives. The basic bond polynomial and cycle polynomial are…”

“…Since there is only a single external momentum q all matrices not contracted with metric tensors are contracted to / q = q ρ γ ρ , and with / q / q = q 2 one finds that one always just has an integer multiple of a power of q 2 . Applying this to the parametric integrand for QED Feynman integrals from [20], which we will do in more detail in section 4, yields sums of the form…”

“…In gauge theories this becomes much more complicated and until recently the integrand could only be expressed in terms of complicated derivatives of the scalar integrand [18,27]. For quantum electrodynamics the combinatorics of these derivatives have been analysed in [20] and it was found that they can be expressed explicitly in terms of graph polynomials similar to Ψ Γ and Φ Γ . The other complication of QED, the tensor structure consisting of products of Dirac matrices, was dealt with in [21].…”