17,197 research outputs found

    Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies

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    I introduce an approximation scheme that allows to deduce differential equations for the renormalization group β\beta-function from a Schwinger--Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess--Zumino model and a ϕ63\phi^3_6 scalar model which do not have divergent vertex functions, this simple Schwinger--Dyson equation for the propagator captures the main quantum corrections.Comment: Clarification of the presentation of results. Equations and results unchanged. Match the published version. 12 page

    New mathematical structures in renormalizable quantum field theories

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    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

    From quantum electrodynamics to posets of planar binary trees

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    This paper is a brief mathematical excursion which starts from quantum electrodynamics and leads to the Moebius function of the Tamari lattice of planar binary trees, within the framework of groups of tree-expanded series. First we recall Brouder's expansion of the photon and the electron Green's functions on planar binary trees, before and after the renormalization. Then we recall the structure of Connes and Kreimer's Hopf algebra of renormalization in the context of planar binary trees, and of their dual group of tree-expanded series. Finally we show that the Moebius function of the Tamari posets of planar binary trees gives rise to a particular series in this group.Comment: 13 page

    A Unified Elementary Approach to the Dyson, Morris, Aomoto, and Forrester Constant Term Identities

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    We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.Comment: 20 page

    Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology

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    In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.Comment: 29 pages, eps figures; minor changes; final versio

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

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    The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar Ď•3\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=4d=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=6d=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

    Disk Partition Function and Oscillatory Rolling Tachyons

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    An exact cubic open string field theory rolling tachyon solution was recently found by Kiermaier et. al. and Schnabl. This oscillatory solution has been argued to be related by a field redefinition to the simple exponential rolling tachyon deformation of boundary conformal theory. In the latter approach, the disk partition function takes a simple form. Out of curiosity, we compute the disk partition function for an oscillatory tachyon profile, and find that the result is nevertheless almost the same.Comment: 17 pages, 2 figures. v4: discussion clarified, appendix added, conclusions unchanged; version to appear in J.Phys.
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