Some combinatorial interpretations in perturbative quantum field theory

Abstract

This paper will describe how combinatorial interpretations can help us understand the algebraic structure of two aspects of perturbative quantum field theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman graphs. The particular examples which will be looked at are, a better reduction to geometric series for Dyson-Schwinger equations, a subgraph which yields extra denominator reductions in scalar Feynman integrals, and an explanation of a trick of Brown and Schnetz to get one extra step in the denominator reduction of an important particular graph.

Forbidden minors for graphs with no first obstruction to parametric Feynman integration

“…and hence that the determinant of M G does not depend on the orderings, orientations, or choice of removed vertex (see Proposition 3.7 of [20] for a concise derivation of this from the standard form of the matrix-tree theorem).…”

c₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction

“…In the context of Legendre sums this technique is based on the elimination of square factors from the Legendre symbol. For classical point-counts scaling was already used in [25] and later in [9], with some observations on some combinatorial conditions which allow it in [29]. In this article we have a case where the result after scaling can be further reduced by additional steps of quadratic denominator reduction.…”

Quantum fields, periods and algebraic geometry

“…A regularized integrand can be obtained by raising the denominator 1/ψ 2 Γ to a noninteger power (dimensional regularization), or multiplication by non-integer powers of edge variables, together with suitable Γ-functions (analytic regularization). The latter suffices to treat the Mellin transforms as used for example in [22] and discussed below.…”

“…For example, with B 3 + now effecting a 1-scale insertion, B 3 + (Γ 4 ) = G 2 and bα 3 (Γ 4 ) = α 3 (I)Γ 4 : Φ R ((B 3 + + bα 3 )(Γ 4 ) = Φ R (Γ 2 ) + 20ζ(5)α 3 (I)L,where we are free to choose α 3 (I) to modify d 1 3 (Θ), a useful fact in light of the manipulations in[21,22].…”