431 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.
Non Local Theories: New Rules for Old Diagrams
We show that a general variant of the Wick theorems can be used to reduce the
time ordered products in the Gell-Mann & Low formula for a certain class on non
local quantum field theories, including the case where the interaction
Lagrangian is defined in terms of twisted products.
The only necessary modification is the replacement of the
Stueckelberg-Feynman propagator by the general propagator (the ``contractor''
of Denk and Schweda)
D(y-y';tau-tau')= - i
(Delta_+(y-y')theta(tau-tau')+Delta_+(y'-y)theta(tau'-tau)), where the
violations of locality and causality are represented by the dependence of
tau,tau' on other points, besides those involved in the contraction. This leads
naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms
of the same diagrams as in the local case, the only necessary modification
concerning the Feynman rules. The ordinary local theory is easily recovered as
a special case, and there is a one-to-one correspondence between the local and
non local contributions corresponding to the same diagrams, which is preserved
while performing the large scale limit of the theory.Comment: LaTeX, 14 pages, 1 figure. Uses hyperref. Symmetry factors added;
minor changes in the expositio
Dimensional renormalization: ladders to rainbows
Renormalization factors are most easily extracted by going to the massless
limit of the quantum field theory and retaining only a single momentum scale.
We derive factors and renormalized Green functions to all orders in
perturbation theory for rainbow graphs and vertex (or scattering diagrams) at
zero momentum transfer, in the context of dimensional renormalization, and we
prove that the correct anomalous dimensions for those processes emerge in the
limit D -> 4.Comment: RevTeX, no figure
The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization
Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf
algebra of renormalization in perturbative quantum field theory, we investigate
the relation between the twisted antipode axiom in that formalism, the Birkhoff
algebraic decomposition and the universal formula of Kontsevich for quantum
deformation.Comment: 21 pages, 15 figure
On the Brownian gas: a field theory with a Poissonian ground state
As a first step towards a successful field theory of Brownian particles in
interaction, we study exactly the non-interacting case, its combinatorics and
its non-linear time-reversal symmetry. Even though the particles do not
interact, the field theory contains an interaction term: the vertex is the
hallmark of the original particle nature of the gas and it enforces the
constraint of a strictly positive density field, as opposed to a Gaussian free
field. We compute exactly all the n-point density correlation functions,
determine non-perturbatively the Poissonian nature of the ground state and
emphasize the futility of any coarse-graining assumption for the derivation of
the field theory. We finally verify explicitly, on the n-point functions, the
fluctuation-dissipation theorem implied by the time-reversal symmetry of the
action.Comment: 31 page
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
Transcendental numbers and the topology of three-loop bubbles
We present a proof that all transcendental numbers that are needed for the
calculation of the master integrals for three-loop vacuum Feynman diagrams can
be obtained by calculating diagrams with an even simpler topology, the topology
of spectacles.Comment: 4 pages in REVTeX, 1 PostScript figure included, submitted to Phys.
Rev. Let
Dimensional Renormalization in phi^3 theory: ladders and rainbows
The sum of all ladder and rainbow diagrams in theory near 6
dimensions leads to self-consistent higher order differential equations in
coordinate space which are not particularly simple for arbitrary dimension D.
We have now succeeded in solving these equations, expressing the results in
terms of generalized hypergeometric functions; the expansion and representation
of these functions can then be used to prove the absence of renormalization
factors which are transcendental for this theory and this topology to all
orders in perturbation theory. The correct anomalous scaling dimensions of the
Green functions are also obtained in the six-dimensional limit.Comment: 11 pages, LaTeX, no figure
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
Operator approach to analytical evaluation of Feynman diagrams
The operator approach to analytical evaluation of multi-loop Feynman diagrams
is proposed. We show that the known analytical methods of evaluation of
massless Feynman integrals, such as the integration by parts method and the
method of "uniqueness" (which is based on the star-triangle relation), can be
drastically simplified by using this operator approach. To demonstrate the
advantages of the operator method of analytical evaluation of multi-loop
Feynman diagrams, we calculate ladder diagrams for the massless theory
(analytical results for these diagrams are expressed in terms of multiple
polylogarithms). It is shown how operator formalism can be applied to
calculation of certain massive Feynman diagrams and investigation of Lipatov
integrable chain model.Comment: 16 pages. To appear in "Physics of Atomic Nuclei" (Proceedings of
SYMPHYS-XII, Yerevan, Armenia, July 03-08, 2006
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