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.

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

In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and small $q^2$, in $d$ 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 $d$, and achieve highly accelerated convergence of its expansions in powers of $q^2/m^2$ or $m^2/q^2$, for $d=4$.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

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

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

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

Small momentum expansion of the "sunset" diagram with three different masses is obtained. Coefficients at powers of $p^2$ 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

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. $m_i\in\{0,m\}$) 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

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 $\phi_6^3$ 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

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

We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $\phi^4$ 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

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