Log expansions from combinatorial Dyson–Schwinger equations

Abstract

We give a precise connection between combinatorial Dyson-Schwinger equations and log-expansions for Green's functions in quantum field theory. The latter are triangular power series in the coupling constant α and a logarithmic energy scale L -a reordering of terms as G(α, L) = 1± j≥0 α j H j (αL) is the corresponding log-expansion. In a first part of this paper, we derive the leading-log order H 0 and the next-to (j) -leading log orders H j from the Callan-Symanzik equation. In particular, H j only depends on the (j + 1)-loop β-function and anomalous dimensions. For the photon propagator Green's function in quantum electrodynamics (and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected), our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e. fixed-point relations for the Green's functions. Combining the arguments, our work provides a derivation of the log-expansions for Green's functions from the corresponding Dyson-Schwinger equations.

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi3 QFT in 6 Dimensions

“…The seminal work of Kreimer and Connes showed that there is an underlying Hopf-algebraic structure to the renormalization of quantum field theory (QFT) [28,29,61]. This new perspective has led to deep insights into QFT, and also to novel computational methods that have enabled significant progress in higher order perturbative computations [14,16,17,20,22,23,27,60,62,63,64,81,83,84,85,88,89,90,91]. The Hopf-algebraic formulation of QFT is inherently perturbative in nature, so an important open question is to understand how the non-perturbative features of QFT arise naturally within the perturbative Hopf algebra structure.…”

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

“…(n + 1) (which is asymptotically equivalent to n k ). Therefore the asymptotic estimate of Theorem 5.2 leads (after simplification) to One naturally recover estimate (10).…”

“…A chord (a, b) in a diagram is terminal if every chord (c, d) intersecting (a, b) satisfies c < a. Equivalently, a chord (a, b) is terminal if there is no chord (a , b ) satisfying a < a < b < b .It means that terminals chords correspond to the vertices with no outgoing edge in the intersection graph of their diagram. For example, chords (4, 6) and(7,10) are terminal in the right diagram ofFigure 1, and chords 4, 6, 7 are terminal in the diagram ofFigure 3.…”

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: $ϕ^3$ QFT in 6 Dimensions

“…The seminal work of Kreimer and Connes showed that there is an underlying Hopf algebraic structure to the renormalization of quantum field theory (QFT) [1][2][3]. This new perspective has led to deep insights into QFT, and also to novel computational methods that have enabled significant progress in higher order perturbative computations [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] The Hopf algebraic formulation of QFT is inherently perturbative in nature, so an important open question is to understand how the nonperturbative features of QFT arise naturally within the perturbative Hopf algebra structure. In a recent paper [22] we showed how this works for 4 dimensional massless Yukawa theory, using Écalle's theory of resurgent trans-series and alien calculus [23][24][25][26][27].…”