414 research outputs found
Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies
I introduce an approximation scheme that allows to deduce differential
equations for the renormalization group -function from a
Schwinger--Dyson equation for the propagator. This approximation is proven to
give the dominant asymptotic behavior of the perturbative solution. In the
supersymmetric Wess--Zumino model and a scalar model which do not
have divergent vertex functions, this simple Schwinger--Dyson equation for the
propagator captures the main quantum corrections.Comment: Clarification of the presentation of results. Equations and results
unchanged. Match the published version. 12 page
Higher loop renormalization of a supersymmetric field theory
Using Dyson--Schwinger equations within an approach developed by Broadhurst
and Kreimer and the renormalization group, we show how high loop order of the
renormalization group coefficients can be efficiently computed in a
supersymmetric model.Comment: 8 pages, 2 figure
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 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
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
On Motives Associated to Graph Polynomials
The appearance of multiple zeta values in anomalous dimensions and
-functions of renormalizable quantum field theories has given evidence
towards a motivic interpretation of these renormalization group functions. In
this paper we start to hunt the motive, restricting our attention to a subclass
of graphs in four dimensional scalar field theory which give scheme independent
contributions to the above functions.Comment: 54
A generic Hopf algebra for quantum statistical mechanics
In this paper, we present a Hopf algebra description of a bosonic quantum
model, using the elementary combinatorial elements of Bell and Stirling
numbers. Our objective in doing this is as follows. Recent studies have
revealed that perturbative quantum field theory (pQFT) displays an astonishing
interplay between analysis (Riemann zeta functions), topology (Knot theory),
combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure).
Since pQFT is an inherently complicated study, so far not exactly solvable and
replete with divergences, the essential simplicity of the relationships between
these areas can be somewhat obscured. The intention here is to display some of
the above-mentioned structures in the context of a simple bosonic quantum
theory, i.e. a quantum theory of non-commuting operators that do not depend on
space-time. The combinatorial properties of these boson creation and
annihilation operators, which is our chosen example, may be described by
graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf
algebra structure. Our approach is based on the quantum canonical partition
function for a boson gas.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text
overlap with arXiv:1011.052
Heavy-Higgs Lifetime at Two Loops
The Standard-Model Higgs boson with mass decays almost
exclusively to pairs of and bosons. We calculate the dominant two-loop
corrections of to the partial widths of these decays. In
the on-mass-shell renormalization scheme, the correction factor is found to be
, where the second term is the
one-loop correction. We give full analytic results for all divergent two-loop
Feynman diagrams. A subset of finite two-loop vertex diagrams is computed to
high precision using numerical techniques. We find agreement with a previous
numerical analysis. The above correction factor is also in line with a recent
lattice calculation.Comment: 26 pages, 6 postscript figures. The complete paper including figures
is also available via WWW at
http://www.physik.tu-muenchen.de/tumphy/d/T30d/PAPERS/TUM-HEP-247-96.ps.g
Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
We describe a unification of several apparently unrelated factorizations
arisen from quantum field theory, vertex operator algebras, combinatorics and
numerical methods in differential equations. The unification is given by a
Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff
formula in our study of the Hopf algebra approach of Connes and Kreimer to
renormalization in perturbative quantum field theory. There we showed that the
Birkhoff decomposition of Connes and Kreimer can be obtained from a certain
Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter
operator. We will explain how the same decomposition generalizes the
factorization of formal exponentials and uniformization for Lie algebras that
arose in vertex operator algebra and conformal field theory, and the even-odd
decomposition of combinatorial Hopf algebra characters as well as to the Lie
algebra polar decomposition as used in the context of the approximation of
matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy
From Quantum Mechanics to Quantum Field Theory: The Hopf route
We show that the combinatorial numbers known as {\em Bell numbers} are
generic in quantum physics. This is because they arise in the procedure known
as {\em Normal ordering} of bosons, a procedure which is involved in the
evaluation of quantum functions such as the canonical partition function of
quantum statistical physics, {\it inter alia}. In fact, we shall show that an
evaluation of the non-interacting partition function for a single boson system
is identical to integrating the {\em exponential generating function} of the
Bell numbers, which is a device for encapsulating a combinatorial sequence in a
single function. We then introduce a remarkable equality, the Dobinski
relation, and use it to indicate why renormalisation is necessary in even the
simplest of perturbation expansions for a partition function. Finally we
introduce a global algebraic description of this simple model, giving a Hopf
algebra, which provides a starting point for extensions to more complex
physical systems
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