Related papers: Non RG logarithms via RG equations
We find the leading RG logs in $\phi^4$ theory for any Feynman diagram with 4 external edges. We obtain the result in two ways. The first way is to calculate the relevant terms in Feynman integrals. The second way is to use the RG…
Connes and Kreimer have discovered a Hopf algebra structure behind renormalization of Feynman integrals. We generalize the Hopf algebra to the case of ribbon graphs, i.e. to the case of theories with matrix fields. The Hopf algebra is…
We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be…
We study the renormalization group equations following from the Hopf algebra of graphs. Vertex functions are treated as vectors in dual to the Hopf algebra space. The RG equations on such vertex functions are equivalent to RG equations on…
The RG equation for the effective potential in the leading log (LL) approximation is constructed which is valid for an arbitrary scalar field theory in 4 dimensions. The solution to this equation sums up the leading $\log\phi$ contributions…
We demonstrate that the fundamental algebraic structure underlying the Connes-Kreimer Hopf algebra -- the insertion pre-Lie structure on graphs -- corresponds directly to the canonical pre-Lie structure of polynomial vector fields. Using…
We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes' and Kreimer's algebraic Birkhoff decomposition to renormalize multiple polylogarithms at…
We apply our previously developed approach to marginal quartic interactions in multiscalar QFTs, which shows that one-loop RG flows can be described in terms of a commutative algebra, to various models in 4d. We show how the algebra can be…
Based on Hopf algebra of rooted trees introduced by Connes and Kreimer, we construct a class of linear maps on noncommutative polynomial algebra in two indeterminates, namely rooted tree maps. We also prove that their maps induce a class of…
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests…
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of…
We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative…
For any root system corresponding to a semisimple simply-laced Lie algebra a logarithmic CFT is constructed. Characters of irreducible representations were calculated in terms of theta functions.
We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series…
We give a universal construction of families of Hopf $P$-algebras for any Hopf operad $P$. As a special case, we recover the Connes-Kreimer Hopf algebra of rooted trees.
Explicit two-loop calculations in noncommutative $\phi^4_4$ theory are presented. It is shown that the model is two-loop renormalizable.
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's…
We introduce a generalization of the scaled relative graph (SRG) to pairs of operators, enabling the visualization of their relative incremental properties. This novel SRG framework provides the geometric counterpart for the study of…
In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the…
We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective…