Non-perturbative β-functions via Feynman graphons

Abstract

We show the existence of a new class of [Formula: see text]-functions which can govern the running of strong coupling constants in gauge field theories.

Halting problem in Feynman graphon processes derived from the renormalization Hopf algebra

“…It is also possible to study random graphs and homomorphism densities for these non-zero graphons. Several applications of graphons in Combinatorics, Theoretical Computer Science and Quantum Field Theory have been addressed in [7,9,11,12,18,24,25,26,27,28].…”

“…This new setting provides a new random graph model for the study of the non-perturbative behavior of quantum equations. [24,25,26,27,28] 2 From Lebesgue to L p Feynman graphons…”

“…The theory of graph functions (or graphons) associated to the space of dense or sparse graphs has been developed in Infinite Combinatorics [4,5,6,7,8,9,10,15] and Computer Science [11,12,18,29]. Recently, some new applications of graphons to Quantum Field Theory are investigated [24,25] where Feynman graphons, as analytic generalizations of Feynman diagrams, have been built to provide a new analytic platform for the study of the non-perturbative behavior of equations of motions in strongly coupled gauge field theories [26,27,28].…”

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

“…In addition, thanks to the topology of graphons, it is also possible to describe the inverse of these Green's functions at infinite loop orders which enables us to provide some new non-perturbative computational methods. For example in [29], we have formulated a class of β-functions which control the behavior of solutions of Dyson-Schwinger equations under changing the scales of running coupling constants.…”

“…This unique solution determines a graded commutative (finite type) Hopf subalgebra H DSE of the renormalization Hopf algebra of Feynman diagrams which is free algebraically generated by objects X n 's [17,29,30,36]. …”