138 research outputs found

    Using the Hopf Algebra Structure of QFT in Calculations

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    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.

    Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension

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    In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and small q2q^2, in dd dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Pad\'{e} approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon self-energy, for all dd, and achieve highly accelerated convergence of its expansions in powers of q2/m2q^2/m^2 or m2/q2m^2/q^2, for d=4d=4.Comment: 25 pages, OUT--4102--43, BI--TP/92--5

    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 Ï•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

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

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    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=q2/4m2z=q^2/4m^2. In the limiting cases z→0z\to0, z→1z\to1, and z→−∞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 ≈0.021\approx0.021 GeV4^4.Comment: 9 page

    On evaluation of two-loop self-energy diagram with three propogator

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    Small momentum expansion of the "sunset" diagram with three different masses is obtained. Coefficients at powers of p2p^2 are evaluated explicitly in terms of dilogarithms and elementary functions. Also some power expansions of "sunset" diagram in terms of different sets of variables are given.Comment: 9 pages, LaTEX, MSU-PHYS-HEP-Lu3/9

    High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles

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    In this article we present a high-precision evaluation of the expansions in \e=(4-d)/2 of (up to) four-loop scalar vacuum master integrals, using the method of difference equations developed by S. Laporta. We cover the complete set of `QED-type' master integrals, i.e. those with a single mass scale only (i.e. mi∈{0,m}m_i\in\{0,m\}) and an even number of massive lines at each vertex. Furthermore, we collect all that is known analytically about four-loop `QED-type' masters, as well as about {\em all} single-mass-scale vacuum integrals at one-, two- and three-loop order.Comment: 25 pages, uses axodraw.st

    An Efficient Method for the Solution of Schwinger--Dyson equations for propagators

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    Efficient computation methods are devised for the perturbative solution of Schwinger--Dyson equations for propagators. We show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess--Zumino model and the Ï•63\phi_6^3 complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.Comment: 9 pages, no figures. Match of the published version, with the corrections in proo

    Unknotting the polarized vacuum of quenched QED

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    A knot-theoretic explanation is given for the rationality of the quenched QED beta function. At the link level, the Ward identity entails cancellation of subdivergences generated by one term of the skein relation, which in turn implies cancellation of knots generated by the other term. In consequence, each bare three-loop diagram has a rational Laurent expansion in the Landau gauge, as is verified by explicit computation. Comparable simplification is found to occur in scalar electrodynamics, when computed in the Duffin-Kemmer-Petiau formalism.Comment: 11 pages, LaTe

    Spanning forest polynomials and the transcendental weight of Feynman graphs

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    We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in Ï•4\phi^4 theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight.Comment: 30 page

    Bjorken unpolarized and polarized sum rules: comparative analysis of large-N_F expansions

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    Analytical all-orders results are presented for the one-renormalon-chain contributions to the Bjorken unpolarized sum rule for the F_1 structure function of nu N deep-inelastic scattering in the large-N_F limit. The feasibility of estimating higher order perturbative QCD corrections, by the process of naive nonabelianization (NNA), is studied, in anticipation of measurement of this sum rule at a Neutrino Factory. A comparison is made with similar estimates obtained for the Bjorken polarized sum rule. Application of the NNA procedure to correlators of quark vector and scalar currents, in the euclidean region, is compared with recent analytical results for the O(alpha_s^4 N_F^2) terms.Comment: 9 page
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