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

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

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.

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

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

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

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

Operator product expansion in static-quark effective field theory: large perturbative correction

We calculate the coefficients of operators with dimensions d <= 7 in the operator product expansion of correlators of q Gamma Q currents, for the effective field theory of an infinite-mass quark, Q. Exact two-loop results are obtained, with an arbitrary gauge group and spacetime dimension, for the perturbative (d=0) and quark-condensate (d=3) contributions, confirming our previous result for the anomalous dimension of the current. Leading-order results are given for light-quark operators with d=5, 6, 7 and gluon operators with d=4, 6. The existence of a perturbative correction of order 100% precludes a reliable determination of f_B from non-relativistic sum rules.Comment: Old preprint (a typo in dimension-7 contribution corrected

Connections between Deep-Inelastic and Annihilation Processes at Next-to-Next-to-Leading Order and Beyond

We have discovered 7 intimate connections between the published results for the radiative corrections, \Ck, to the Gross--Llewellyn Smith (GLS) sum rule, in deep-inelastic lepton scattering, and the radiative corrections, \Cr, to the Adler function of the flavour-singlet vector current, in \ee annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED $\beta$\/-function and the zero-fermion-loop abelian terms in the 3-loop GLS sum rule. The combined effect of all 7 relations is to give the factorization of the 2-loop $\beta$\/-function in \Ds\equiv\Ck\Cr-1=\frac{\Be}{\Aq}\left\{S_1\Cf\Aq+\left[S_2\Tf\Nf +\Sa\Ca+\Sf\Cf\right]\Cf\Aq^2\right\}+O(\Aq^4)\,, where \Aq=\al(\mu^2=Q^2)/4\pi is the \MS coupling of an arbitrary colour gauge theory, and S_1=-\Df{21}{2}+12\Ze3\,;\quad S_2=\Df{326}{3}-\Df{304}{3}\Ze3\,;\quad \Sa=-\Df{629}{2}+\Df{884}{3}\Ze3\,;\quad \Sf=\Df{397}{6}+136\Ze3-240\Ze5 specify the sole content of \Ck that is not already encoded in \Cr and \Be=Q^2\rd\Aq/\rd Q^2 at O(\Aq^3). The same result is obtained by combining the radiative corrections to Bjorken's polarized sum rule with those for the Adler function of the non-singlet axial current. We suggest possible origins of $\beta$ in the `Crewther discrepancy', \Ds, and determine \Ds/(\Be/\Aq), to all orders in \Nf\Aq, in the large-\Nf limit, obtaining the {\em entire\/} series of coefficients of which $S_1$ and $S_2$ are merely the first two members.Comment: 11 pages, LATEX, preprint INR-820/93, OUT-4102-45; In memoriam Sergei Grogorievich Corishny, 1958-198