351 research outputs found
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
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
Loop Integrals, R Functions and their Analytic Continuation
To entirely determine the resulting functions of one-loop integrals it is
necessary to find the correct analytic continuation to all relevant kinematical
regions. We argue that this continuation procedure may be performed in a
general and mathematical accurate way by using the function notation
of these integrals. The two- and three-point cases are discussed explicitly in
this manner.Comment: 10 pages (Latex), MZ-TH/93-1
Polynomial functors and combinatorial Dyson-Schwinger equations
We present a general abstract framework for combinatorial Dyson-Schwinger
equations, in which combinatorial identities are lifted to explicit bijections
of sets, and more generally equivalences of groupoids. Key features of
combinatorial Dyson-Schwinger equations are revealed to follow from general
categorical constructions and universal properties. Rather than beginning with
an equation inside a given Hopf algebra and referring to given Hochschild
-cocycles, our starting point is an abstract fixpoint equation in groupoids,
shown canonically to generate all the algebraic structure. Precisely, for any
finitary polynomial endofunctor defined over groupoids, the system of
combinatorial Dyson-Schwinger equations has a universal solution,
namely the groupoid of -trees. The isoclasses of -trees generate
naturally a Connes-Kreimer-like bialgebra, in which the abstract
Dyson-Schwinger equation can be internalised in terms of canonical
-operators. The solution to this equation is a series (the Green function)
which always enjoys a Fa\`a di Bruno formula, and hence generates a
sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying yields
different bialgebras, and cartesian natural transformations between various
yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to
truncation of Dyson-Schwinger equations. Finally, all constructions can be
pushed inside the classical Connes-Kreimer Hopf algebra of trees by the
operation of taking core of -trees. A byproduct of the theory is an
interpretation of combinatorial Green functions as inductive data types in the
sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy
Knots and Numbers in Theory to 7 Loops and Beyond
We evaluate all the primitive divergences contributing to the 7--loop
\/--function of theory, i.e.\ all 59 diagrams that are free of
subdivergences and hence give scheme--independent contributions. Guided by the
association of diagrams with knots, we obtain analytical results for 56
diagrams. The remaining three diagrams, associated with the knots ,
, and , are evaluated numerically, to 10 sf. Only one
satellite knot with 11 crossings is encountered and the transcendental number
associated with it is found. Thus we achieve an analytical result for the
6--loop contributions, and a numerical result at 7 loops that is accurate to
one part in . The series of `zig--zag' counterterms,
,
previously known for loops, is evaluated to 10 loops, corresponding
to 17 crossings, revealing that the \/--loop zig--zag term is , where are the Catalan numbers, familiar in knot theory. The investigations
reported here entailed intensive use of REDUCE, to generate
lines of code for multiple precision FORTRAN computations, enabled by Bailey's
MPFUN routines, running for CPUhours on DecAlpha machines.Comment: 6 pages plain LaTe
Renormalization of gauge fields using Hopf algebras
We describe the Hopf algebraic structure of Feynman graphs for non-abelian
gauge theories, and prove compatibility of the so-called Slavnov-Taylor
identities with the coproduct. When these identities are taken into account,
the coproduct closes on the Green's functions, which thus generate a Hopf
subalgebra.Comment: 16 pages, 1 figure; uses feynmp. To appear in "Recent Developments in
Quantum Field Theory". Eds. B. Fauser, J. Tolksdorf and E. Zeidler.
Birkhauser Verlag, Basel 200
Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
In this article we continue to explore the notion of Rota-Baxter algebras in
the context of the Hopf algebraic approach to renormalization theory in
perturbative quantum field theory. We show in very simple algebraic terms that
the solutions of the recursively defined formulae for the Birkhoff
factorization of regularized Hopf algebra characters, i.e. Feynman rules,
naturally give a non-commutative generalization of the well-known Spitzer's
identity. The underlying abstract algebraic structure is analyzed in terms of
complete filtered Rota-Baxter algebras.Comment: 19 pages, 2 figure
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