Graphons and renormalization of large Feynman diagrams

Abstract

Abstract. The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions.

Non-perturbative β-functions via Feynman graphons

“…For a given Dyson-Schwinger equation DSE with the corresponding sequence {Y m } m≥1 of partial sums, which is convergent to the unique solution X DSE with respect to the cutdistance topology [19], the sequence {Sφ Rms ([W Ym ])} m≥1 is also convergent where thanks to the continuity of the twisted antipode we have…”

“…If we work on the geometric setting for the study of Dyson-Schiwnger equations built in [16,17,18] and also apply their graphon representation given in [19], then it is possible to show thatβ can determine a unique class ωβ of flat equi-singular G Φ graphon (C)-connections on the trivial principal bundle P 0 graphon := ∆ × C * − π −1 ({0}) × G Φ graphon (C) derived from the regularization process. This correspondence can be presented in terms of the differential equation…”

“…The rooted tree representations of Feynman diagrams in a given sequence of these graphs have been applied to associate a new class of dense graphs with respect to the graph limits of Feynman diagrams. This method provided a new description for solutions of Dyson-Schwinger equations which has led us to improve the formulation of a renormalization machinery for these non-perturbative type of equations [19].…”

Formal aspects of non-perturbative Quantum Field Theory via an operator theoretic setting

“…The pixel picture presentations of rooted trees can be applied to translate the topology of graphons to rooted trees and equip H CK (I) with the cut-distance topology to obtain a compact topological Hopf algebra denoted by H cut CK (I). It is then shown that X is the cut-distance convergent limit of the sequence {X (n) } n≥0 of its partial sums [30].…”

“…Furthermore, the geometric interpretation of the modified Standard Model has shown the essential role of Noncommutative Geometry and its potential for the description of modern physical theories. [2,3,17,18,21,22,31,30,35] In Quantum Field Theories with strong couplings, we need to deal with even more complicated problems originated from infinite formal expansions of Feynman integrals (or Feynman diagrams) under running and bare coupling constants. Fixed point equations of Green's functions are the original tools to classify these expansions where we study Dyson-Schwinger equations.…”