Geometrically Relating Momentum Cut-Off and Dimensional Regularization

Abstract

The β function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the β functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories. β function, Hopf Algebra, momentum cut-off regularization, dimensional regularization 1

Dynkin operators, renormalization and the geometric $β$ function

“…The β function for dimensional regularization and momentum cutoff regularization satisfies an equation of the form (1). In [6,2], the authors show that this β function defines a connection that also satisfies (1). In this note, I show that there is a much deeper connection between the Dynkin operator and renormalization.…”

“…In [5], the authors define a β function, a Lie algebra element representing how a dimensionally regularized QFT depends on the energy scale. The β function for dimensional regularization and momentum cutoff regularization satisfies an equation of the form (1). In [6,2], the authors show that this β function defines a connection that also satisfies (1).…”

A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory

“…This setting has been lifted onto a uni-versal Tannakian formalism where a renormalizable Quantum Field Theory is studied via a category of geometric objects which can be recovered by a category of finite dimensional representations of the affine group scheme G Φ := Hom(H FG (Φ), −). [2,3,16,18,19,20,21,24,22,26,33,34,37,65,66,87,95,96,97,98,125,155,156,168,174] Perhaps the most fundamental result in this direction would be the discovery of a very deep interrelationship between Feynman integrals and theory of motives in Algebraic Geometry where a motivic renormalization machinery has been formulated to deal with divergencies in the language of Picard-Fuchs equations and other powerful tools.…”

“…It is easy to check that for a given measure preserving map ρ, the map ρ ⊗ ρ : Ω 2 1 −→ Ω 2 2 defined by ρ ⊗ ρ(x, y) := (ρ(x), ρ(y)) is also a measure preserving map. If ρ is a bijection, then f ρ , W ρ are called rearrangements of f (as a function on Ω 2 ) and W (as a function on Ω 2 2 ). Actually, relabeling of labeled graphons can be understood as a kind of rearrangement.…”