Filtrations in Dyson-Schwinger equations: next-to^j -leading log\n expansions systematically

2014/12/04 by Olaf Krueger, Dirk Kreimer, Krueger, Olaf +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Database Systems and Queries #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Mechanics and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1412.1657

openalex publication_date 2014/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Dyson-Schwinger equations determine the Green functions G^r(\α,L) in\nquantum field theory. Their solutions are triangular series in a coupling\nconstant \α and an external scale parameter L for a chosen amplitude\nr, with the order in L bounded by the order in the coupling. Perturbation\ntheory calculates the first few orders in \α. On the other hand,\nDyson--Schwinger equations determine next-to^ \j -leading log\nexpansions, G^r(\α,L) = 1 + \∑_j=0^\∞ \∑_\M\np_j^\M\α^j \M(u). \∑_\M sums a finite\nnumber of functions \M in u = \α L/2. The leading logs come\nfrom the trivial representation \M(u) =\n beginbsmallmatrix bullet endbsmallmatrix(u) at j=0 with\np₀^ beginbsmallmatrix bullet endbsmallmatrix = 1. All non-leading logs\nare organized by the suppression in powers \α^j. We describe an algebraic\nmethod to derive all next-to^ \j -leading log terms from the\nknowledge of the first (j+1) terms in perturbation theory and their\nfiltrations. This implies the calculation of the functions \M(u) and\nperiods p_j^\M. In the first part of our paper, we investigate the\nstructure of Dyson-Schwinger equations and develop a method to filter their\nsolutions. Applying renormalized Feynman rules maps each filtered term to a\ncertain power of \α and L in the log-expansion. Based on this, the\nsecond part derives the next-to^ \j -leading log expansions. Our\nmethod is general. Here, we exemplify it using the examples of the propagator\nin Yukawa theory and the photon self-energy in quantum electrodynamics. The\nreader may apply our method to any (set of) Dyson-Schwinger equation(s)\nappearing in renormalizable quantum field theories.\n

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