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.

New mathematical structures in renormalizable quantum field theories

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos correcte

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

Multiloop calculations in HQET

Recently, algorithms for calculation of 3-loop propagator diagrams in HQET and on-shell QCD with a heavy quark have been constructed and implemented. These algorithms (based on integration by parts recurrence relations) reduce an arbitrary diagram to a combination of a finite number of basis integrals. Here I discuss various ways to calculate non-trivial bases integrals, either exactly or as expansions in epsilon. Some integrals of these two classes are related to each other by inversion, which provides a useful cross-check.Comment: Talk at ACAT'2002 (Moscow

Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension $d=4$, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at $d=6$, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.Comment: 20p, 2 epsf fi

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

Interaction, reaction and performance: The human body tracking project

As a result of new technological advancements in performance practice, I am arguing that new liminal spaces exist where there is a potential for a reconfiguration of creativity and experimentation in performance practice