611 research outputs found
Unique factorization in perturbative QFT
We discuss factorization of the Dyson--Schwinger equations using the Lie- and
Hopf algebra of graphs. The structure of those equations allows to introduce a
commutative associative product on 1PI graphs. In scalar field theories, this
product vanishes if and only if one of the factors vanishes. Gauge theories are
more subtle: integrality relates to gauge symmetries.Comment: 5pages, Talk given at "RadCor 2002 - Loops and Legs 2002", Kloster
Banz, Germany, Sep 8-13, 200
What is the trouble with Dyson--Schwinger equations?
We discuss similarities and differences between Green Functions in Quantum
Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint
equations which originate from an underlying Hopf algebra structure. Typically,
the equation is linear for the polylog, and non-linear for Green Functions. We
argue though that the crucial difference lies not in the non-linearity of the
latter, but in the appearance of non-trivial representation theory related to
transcendental extensions of the number field which governs the linear
solution. An example is studied to illuminate this point.Comment: 5 pages contributed to the proceedings "Loops and Legs 2004", April
2004, Zinnowitz, German
Renormalization automated by Hopf algebra
It was recently shown that the renormalization of quantum field theory is
organized by the Hopf algebra of decorated rooted trees, whose coproduct
identifies the divergences requiring subtraction and whose antipode achieves
this. We automate this process in a few lines of recursive symbolic code, which
deliver a finite renormalized expression for any Feynman diagram. We thus
verify a representation of the operator product expansion, which generalizes
Chen's lemma for iterated integrals. The subset of diagrams whose forest
structure entails a unique primitive subdivergence provides a representation of
the Hopf algebra of undecorated rooted trees. Our undecorated Hopf
algebra program is designed to process the 24,213,878 BPHZ contributions to the
renormalization of 7,813 diagrams, with up to 12 loops. We consider 10 models,
each in 9 renormalization schemes. The two simplest models reveal a notable
feature of the subalgebra of Connes and Moscovici, corresponding to the
commutative part of the Hopf algebra of the diffeomorphism group:
it assigns to Feynman diagrams those weights which remove zeta values from the
counterterms of the minimal subtraction scheme. We devise a fast algorithm for
these weights, whose squares are summed with a permutation factor, to give
rational counterterms.Comment: 22 pages, latex, epsf for figure
Feynman diagrams as a weight system: four-loop test of a four-term relation
At four loops there first occurs a test of the four-term relation derived by
the second author in the course of investigating whether counterterms from
subdivergence-free diagrams form a weight system. This test relates
counterterms in a four-dimensional field theory with Yukawa and
interactions, where no such relation was previously suspected. Using
integration by parts, we reduce each counterterm to massless two-loop two-point
integrals. The four-term relation is verified, with , demonstrating non-trivial cancellation of
the trefoil knot and thus supporting the emerging connection between knots and
counterterms, via transcendental numbers assigned by four-dimensional field
theories to chord diagrams. Restrictions to scalar couplings and renormalizable
interactions are found to be necessary for the existence of a pure four-term
relation. Strong indications of richer structure are given at five loops.Comment: minor changes, references updated, 10 pages, LaTe
Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries
We discuss the prominence of Hopf algebras in recent progress in Quantum
Field Theory. In particular, we will consider the Hopf algebra of
renormalization, whose antipode turned out to be the key to a conceptual
understanding of the subtraction procedure. We shall then describe several
occurences of this or closely related Hopf algebras in other mathematical
domains, such as foliations, Runge Kutta methods, iterated integrals and
multiple zeta values. We emphasize the unifying role which the Butcher group,
discovered in the study of numerical integration of ordinary differential
equations, plays in QFT.Comment: Survey paper, 12 pages, epsf for figures, dedicated to Mosh\'e Flato,
minor corrections, to appear in Lett.Math.Phys.4
A new Method for Computing One-Loop Integrals
We present a new program package for calculating one-loop Feynman integrals,
based on a new method avoiding Feynman parametrization and the contraction due
to Passarino and Veltman. The package is calculating one-, two- and three-point
functions both algebraically and numerically to all tensor cases. This program
is written as a package for Maple. An additional Mathematica version is planned
later.Comment: 12 pages Late
Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
The renormalization of quantum field theory twists the antipode of a
noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of
primitive divergences. The Hopf algebra of undecorated rooted trees, , generated by a single primitive divergence, solves a universal problem
in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the
cocommutative subalgebra of pure ladder diagrams and
the Connes-Moscovici noncocommutative subalgebra of
noncommutative geometry. These three Hopf algebras admit a bigrading by ,
the number of nodes, and an index that specifies the degree of primitivity.
In each case, we use iterations of the relevant coproduct to compute the
dimensions of subspaces with modest values of and and infer a simple
generating procedure for the remainder. The results for
are familiar from the theory of partitions, while those for
involve novel transforms of partitions. Most beautiful is the bigrading of
, the largest of the three. Thanks to Sloane's {\tt superseeker},
we discovered that it saturates all possible inequalities. We prove this by
using the universal Hochschild-closed one-cocycle , which plugs one set of
divergences into another, and by generalizing the concept of natural growth
beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater
challenge of handling the infinite set of decorations of realistic quantum
field theory.Comment: 21 pages, LaTe
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
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 , we
obtain an integrodifferential Dyson-Schwinger equation and solve it
parametrically in terms of the complementary error function. For the scalar
theory, at , 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
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