Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

It is found that the number, $M_n$, of irreducible multiple zeta values (MZVs) of weight $n$, is generated by $1-x^2-x^3=\prod_n (1-x^n)^{M_n}$. For $9\ge n\ge3$, $M_n$ enumerates positive knots with $n$ crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops.Comment: 15 pages, Latex, figures using EPSF, replaced version has references and conclusions updated, Eq.(7) revised; as to appear in Phys.Lett.

Unique factorization in perturbative QFT

We discuss factorization of the Dyson--Schwinger equations using the Lie- and Hopf algebra of graphs. The structure of those equations allows to introduce a commutative associative product on 1PI graphs. In scalar field theories, this product vanishes if and only if one of the factors vanishes. Gauge theories are more subtle: integrality relates to gauge symmetries.Comment: 5pages, Talk given at "RadCor 2002 - Loops and Legs 2002", Kloster Banz, Germany, Sep 8-13, 200

Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations

The coefficient functions of the gluon condensate $$, in the correlators of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained analytically, to two loops, for all values of $z=q^2/4m^2$. In the limiting cases $z\to0$, $z\to1$, and $z\to-\infty$, comparisons are made with previous partial results. Approximation methods, based on these limiting cases, are critically assessed, with a view to three-loop work. High accuracy is achieved using a few moments as input. A {\em single} moment, combined with only the {\em leading} threshold and asymptotic behaviours, gives the two-loop corrections to better than 1% in the next 10 moments. A two-loop fit to vector data yields $\approx0.021$ GeV$^4$.Comment: 9 page

Compact analytical form for non-zeta terms in critical exponents at order 1/N^3

We simplify, to a single integral of dilogarithms, the least tractable O(1/N^3) contribution to the large-N critical exponent $\eta$ of the non-linear sigma-model, and hence $\phi^4$-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in epsilon-expansions with $D=2-2\epsilon$, for the sigma-model, and $D=4-2\epsilon$, for $\phi^4$-theory. In both cases we confirm results of Broadhurst, Gracey and Kreimer (BGK) that relate knots to counterterms. The new compact form is much simpler than that of BGK. It enables us to develop 8 new terms in the epsilon-expansion with $D=3-2\epsilon$. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massless Feynman diagrams in odd spacetime dimensions share the greater transcendental complexity of massive diagrams in even dimensions, such as those contributing to the electron's magnetic moment and the electroweak $\rho$-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.Comment: 9 pages, LaTe

Against all odds : the life and music of Michel Petrucciani : a thesis presented in fulfilment of the requirements for the degree of Master of Philosophy at Massey University

Michel Petrucciani, jazz pianist and composer, was a major figure in the history of French jazz who achieved much critical and popular success in his short life span. In 1999 when he died at the age of 36, he was enough of a hero in his own country to warrant inclusion alongside Duke Ellington and Miles Davis in a series of postage stamps celebrating great jazz artists. He was one of only a handful of European jazz musicians to achieve success in the U.S. This thesis examines Petrucciani's life and music with particular emphasis placed on the evolution of his playing and composing styles through close analysis of selected piano solos and a range of compositions representing different periods of his career. An overview of his musical and personal life, as he battled with osteogenesis imperfecta, a rare bone disease that prevents growth to adult size, is also included, along with a study of the influence of pianist Bill Evans on Petrucciani's playing