Shuffle relations for regularised integrals of symbols

We prove shuffle relations which relate a product of regularised integrals of classical symbols to regularised nested (Chen) iterated integrals, which hold if all the symbols involved have non-vanishing residue. This is true in particular for non-integer order symbols. In general the shuffle relations hold up to finite parts of corrective terms arising from renormalisation on tensor products of classical symbols, a procedure adapted from renormalisation procedures on Feynman diagrams familiar to physicists. We relate the shuffle relations for regularised integrals of symbols with shuffle relations for multizeta functions adapting the above constructions to the case of symbols on the unit circle.Comment: 40 pages,latex. Changes concern sections 4 and 5 : an error in section 4 has been corrected, and the link between section 5 and the previous ones has been precise

Field diffeomorphisms and the algebraic structure of perturbative expansions

We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.Comment: 8 pages, 2 figure

Using the Hopf Algebra Structure of QFT in Calculations

We employ the recently discovered Hopf algebra structure underlying perturbative Quantum Field Theory to derive iterated integral representations for Feynman diagrams. We give two applications: to massless Yukawa theory and quantum electrodynamics in four dimensions.Comment: 28 p, Revtex, epsf for figures, minor changes, to appear in Phys.Rev.

Dimensional renormalization: ladders to rainbows

Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4.Comment: RevTeX, no figure

A generic Hopf algebra for quantum statistical mechanics

In this paper, we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is as follows. Recent studies have revealed that perturbative quantum field theory (pQFT) displays an astonishing interplay between analysis (Riemann zeta functions), topology (Knot theory), combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, so far not exactly solvable and replete with divergences, the essential simplicity of the relationships between these areas can be somewhat obscured. The intention here is to display some of the above-mentioned structures in the context of a simple bosonic quantum theory, i.e. a quantum theory of non-commuting operators that do not depend on space-time. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf algebra structure. Our approach is based on the quantum canonical partition function for a boson gas.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text overlap with arXiv:1011.052

Heavy-Higgs Lifetime at Two Loops

The Standard-Model Higgs boson with mass $M_H >> 2M_Z$ decays almost exclusively to pairs of $W$ and $Z$ bosons. We calculate the dominant two-loop corrections of $O( G_F^2 M_H^4 )$ to the partial widths of these decays. In the on-mass-shell renormalization scheme, the correction factor is found to be $1 + 14.6$, where the second term is the one-loop correction. We give full analytic results for all divergent two-loop Feynman diagrams. A subset of finite two-loop vertex diagrams is computed to high precision using numerical techniques. We find agreement with a previous numerical analysis. The above correction factor is also in line with a recent lattice calculation.Comment: 26 pages, 6 postscript figures. The complete paper including figures is also available via WWW at http://www.physik.tu-muenchen.de/tumphy/d/T30d/PAPERS/TUM-HEP-247-96.ps.g

The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.Comment: 21 pages, 15 figure

Feynman graphs, rooted trees, and Ringel-Hall algebras

We construct symmetric monoidal categories \LRF, \FD of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of \LRF, \FD, \HH_{\LRF}, \HH_{\FD} are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman graphs. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable $\phi^4_4$ quantum field theory on the Moyal non commutative ${\mathbb R^4}$ space. This representation involves new {\it hyperbolic} polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information.Comment: 31 pages,10 figure