The GLOBAL Β-Functions FROM SOLUTIONS OF DYSON–SCHWINGER EQUATIONS

Abstract

We apply the geometric interpretation of Dyson-Schwinger equations (DSEs) in terms of equi-singular flat connections to provide a process which relates β-functions of a DSE under different regularization schemes.

Graphons and renormalization of large Feynman diagrams

“…It means that H d FG (Φ) is an enrichment of the renormalization Hopf algebra. This completion will help us improve our previous efforts about the construction of a renormalization program on Dyson-Schwinger equations under an algebro-geometric setting [33,35]. Infinite graphs, which live in the boundary of this complete space, are actually capable to encode solutions of Dyson-Schwinger equations.…”

“…Each term X n is produced on the basis of the terms with the lower degrees and furthermore, terms X n s play the role of generators for a Hopf sub-algebra associated to the equation DSE [1,18,19,25,26,40,41]. This presentation of the solution X, which belongs to the completion of H [[α]] with respect to the n-adic topology, has been applied as a starting point to build some new mathematical structures which are capable to encode non-perturbative parameters [33,35,36]. …”

“…Our best chance in this situation is to make stronger the mathematical foundations of non-perturbative QFT. In this direction, search for some new advanced mathematical structures originated from Dyson-Schwinger equations could be helpful for the better understanding of the phenomenology of non-perturbative parameters [33,35,36,40,41].…”

Non-perturbative β-functions via Feynman graphons

“…Search for the description of the phenomenology of running couplings in QCD, as a top priority in High Energy Physics, can improve our knowledge for the computation and the analysis of non-perturbative parameters. [1,5,6,7,8,15,16] Thanks to the applications of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams to Quantum Field Theory together with other mathematical tools, we have a combinatorial reformulation for (non-linear) Dyson-Schwinger equations in the language of Hochschild cohomology theory. The linear version of these equations can be appeared in the limit of a vanishing β-function.…”

The one-loop θ2 corrections of gauge boson propagator in the noncommutative scalar U(1) theory

“…To overcome the problem, the θ-expanded approach has recently been proposed by many authors and is applied to the renormalizability research of NC gauge theory. [10][11][12][13][14][15][16][17][18][19][20][21] It is indicated that for the pure U(1) NC gauge theory, the propagator of gauge boson is renormalizable for all order of θ at one loop, 10 where the degrees of freedom of SWM provide additional terms to absorb the divergences. For the NC SU(N) pure gauge theory, the gauge propagator is proved to be one-loop renormalizable at θ-order.…”