138 research outputs found

### 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

- …