On matrix differential equations in the Hopf algebra of renormalization

Abstract

We establish Sakakibara's differential equations [M. Sakakibara, On the differential equations of the characters for the renormalization group, Mod. Phys. Lett. A 19 (2004), 1453.] in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.

Geometrically Relating Momentum Cut-Off and Dimensional Regularization

“…It remains to check that β σ defines a one to one correspondence between G(A ) and g(A ) for each σ. This is just an extension of the proof of the similar statement in [9,1].…”

The combinatorics of Bogoliubov’s recursion in renormalization

“…In other words, the matrix of Ψ J [f ] is given by f (M ) := f (M ij ) i,j∈I . It is shown in [14] and also [21] that the map Ψ J defined above is an algebra homomorphism. Its transpose does not depend on the choice of the basis.…”

“…Although we won't detail this point, let us mention that this matrix approach is particularly wellsuited for the study of the renormalization group and the beta-function for local characters in connected graded Hopf algebras with values into meromorphic functions [9]. See [21] as well as [20,14] for a detailed account and applications. 4.1.…”

A Perspective on Regularization and Curvature

“…This is only well defined when γ(z) satisfies condition (4). To find the derivation of the geometric β-function in this context, see [4], [5] or [6].…”

“…Section two of this paper reviews the development of the tools necessary for the construction of the renormalization bundle, following [3], [5], [6]. Section 3 discusses the physical and geometrical β-function, following [15] and [5].…”