40 research outputs found
Summer Student Session 2012 (3/3)
No full text11.15 Michael Borinsky:
Low and High Performance Computing in HEP
11.30 Yasin Buyukalp:
The Automated Control of the ATLAS Pixel Sensor LASER Setu
Tropical Feynman integration in the Minkowski regime
Get PDFWe present a new computer program, feyntrop, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal prescription concretely while retaining projective invariance. feyntrop can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of feyntrop and discuss many explicit examples of evaluated Feynman integrals
Algebraic structure and asymptotics
Get PDFInhalt dieser Arbeit ist eine Erweiterung der Hopfalgebrastruktur der Feynmangraphen und Renormierung von Connes und Kreimer. Zusätzlich wird eine Struktur auf faktoriell wachsenden Potenzreihen eingeführt, die deren asymptotisches Wachstum beschreibt und die kompatibel mit der Hopfalgebrastruktur ist.
Die Hopfalgebrastruktur auf Graphen erlaubt die explizite Enumeration von Graphen mit Einschränkungen in Bezug auf die erlaubten Untergraphen. Im Fall der Feynmangraphen wird zusätzlich eine algebraische Verbandstruktur eingeführt, die weitere eindeutige Eigenschaften von physikalischen Quantenfeldtheorien aufdeckt. Der Differenzialring der faktoriell divergenten Potenzreihen erlaubt es asymptotische Resultate von implizit definierten Potenzreihen mit verschwindendem Konvergenzradius zu extrahieren. In Kombination ergeben beide Strukturen eine algebraische Formulierung großer Graphen mit Einschränkungen für die erlaubten Untergraphen. Diese Strukturen sind motiviert von null-dimensionaler Quantenfeldtheorie and werden zur Analyse ebendieser benutzt.
Als reine Anwendung der Hopfalgebrastruktur wird eine hopfalgebraische Formulierung der Legendretransformation in Quantenfeldtheorien formuliert. Der Differenzialring der faktoriell divergenten Potenzreihen wird dazu benutzt zwei asymptotische Enumerationsprobleme zu lösen: Die asymptotische Anzahl der verbundenen Chorddiagramme und die asymptotische Anzahl der simplen Permutationen. Für beide asymptotischen Lösungen werden vollständige asymptotische Entwicklungen in Form von geschlossenen Erzeugendenfunktionen berechnet. Kombiniert werden beide Strukturen zur Anwendung an null-dimensionaler Quantenfeldtheorie. Zahlreiche Größen werden in den null-dimensionalen Varianten von phi^3, phi^4, QED, quenched QED and Yukawatheorie mit ihren kompletten asymptotischen Entwicklungen berechnet.This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, will be introduced.
The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories.
The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs.
These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory.
As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given.
The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems in combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of phi^3, phi^4, QED, quenched QED and Yukawa theory with their all order asymptotic expansions
Resonant resurgent asymptotics from quantum field theory
Get PDFWe perform an all-order resurgence analysis of a quantum field theory
renormalon that contributes to an anomalous dimension in six-dimensional scalar
theory and is governed by a third-order nonlinear differential
equation. We augment the factorially divergent perturbative expansion
associated to the renormalon by asymptotic expansions to all instanton orders,
in a conjectured and well-tested formula. A distinctive feature of this
renormalon singularity is the appearance of logarithmic terms, starting at
second-instanton order in the trans-series. To highlight this and to illustrate
our methods, we also analyze the trans-series for a closely related
second-order nonlinear differential equation that exhibits a similarly resonant
structure but lacks logarithmic contributions.Comment: 34 pages, v2: Discussion on ODE ambiguities and typos corrected.
Accepted version to appear in Nuclear Physics
Taming a resurgent ultra-violet renormalon
Get PDFPerturbative expansions in quantum field theory diverge for at least two
reasons: the number of Feynman diagrams increases dramatically with the loop
number and the process of renormalization may make the contribution of some
diagrams large. We give an example of the second problem, from an ultra-violent
renormalon of theory in 6 dimensions, where we can compute to very
high loop-order. Taming this renormalon involves recent work on resurgence.
This challenge is much more demanding than the corresponding problem for Yukawa
theory in 4 dimensions.Comment: 8 pages, conference proceedings contribution to Loops and Legs in
Quantum Field Theory 2022 - version to appear in PoS-LL202
