Mesonic chiral perturbation theory - odd intrinsic parity sector

In der vorliegenden Dissertation werden zwei verschiedene Aspekte des Sektors ungerader innerer Parität der mesonischen chiralen Störungstheorie (mesonische ChPT) untersucht. Als erstes wird die Ein-Schleifen-Renormierung des führenden Terms, der sog. Wess-Zumino-Witten-Wirkung, durchgeführt. Dazu muß zunächst der gesamte Ein-Schleifen-Anteil der Theorie mittels Sattelpunkt-Methode extrahiert werden. Im Anschluß isoliert man alle singulären Ein-Schleifen-Strukturen im Rahmen der Heat-Kernel-Technik. Zu guter Letzt müssen diese divergenten Anteile absorbiert werden. Dazu benötigt man eine allgemeinste anomale Lagrange-Dichte der Ordnung O(p^6), welche systematisch entwickelt wird. Erweitert man die chirale Gruppe SU(n)_L x SU(n)_R auf SU(n)_L x SU(n)_R x U(1)_V, so kommen zusätzliche Monome ins Spiel. Die renormierten Koeffizienten dieser Lagrange-Dichte, die Niederenergiekonstanten (LECs), sind zunächst freie Parameter der Theorie, die individuell fixiert werden müssen. Unter Betrachtung eines komplementären vektormesonischen Modells können die Amplituden geeigneter Prozesse bestimmt und durch Vergleich mit den Ergebnissen der mesonischen ChPT eine numerische Abschätzung einiger LECs vorgenommen werden. Im zweiten Teil wird eine konsistente Ein-Schleifen-Rechnung für den anomalen Prozeß (virtuelles) Photon + geladenes Kaon -> geladenes Kaon + neutrales Pion durchgeführt. Zur Kontrolle unserer Resultate wird eine bereits vorhandene Rechnung zur Reaktion (virtuelles) Photon + geladenes Pion -> geladenes Pion + neutrales Pion reproduziert. Unter Einbeziehung der abgeschätzten Werte der jeweiligen LECs können die zugehörigen hadronischen Strukturfunktionen numerisch bestimmt und diskutiert werden.

Lagrange-Singularitäten

In dieser Arbeit wird eine Deformationstheorie fürLagrange-Singularitäten entwickelt. Wir definieren einen Komplex von Moduln mit nicht-linearem Differential, densogenannten Lagrange-de Rham-Komplex, dessen ersteKohomologie isomorph zum Raum der infinitesimalenLagrange-Deformationen ist. Wir beschreiben die Beziehung diesesKomplexes zur Theorie der Moduln über dem Ring vonDifferentieloperatoren. Informationen zur Obstruktionstheorie vonLagrange-Deformationen werden aus derzweiten Kohomologie des Lagrange-de Rham-Komplexes gewonnen.Wir zeigen, dass unter einer geometrischen Bedingung an dieSingularität ie Kohomologie von des Lagrange-deRham-Komplexes ausendlich dimensionalen Vektorräumen besteht. Desweiteren wirdeine Methode zur effektiven Berechnung dieser Kohomologie fürquasi-homogene Lagrange-Flächensingularitäten entwickelt. UnterZuhilfenahme von Computeralgebra wird diese Methode für konkreteBeispiele angewendet.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Static analysis of functionally graded cylindrical and conical shells or panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded conical and cylindrical shells subjected to mechanical loadings. Several types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the conical or cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally conical and cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.

Modelling and analysis of thin-walled beams in the context of the Generalized Beam Theory

In this work, the Generalized Beam Theory (GBT) is used as the main tool to analyze the mechanics of thin-walled beams. After an introduction to the subject and a quick review of some of the most well-known approaches to describe the behaviour of thin-walled beams, a novel formulation of the GBT is presented. This formulation contains the classic shear-deformable GBT available in the literature and contributes an additional description of cross-section warping that is variable along the wall thickness besides along the wall midline. Shear deformation is introduced in such a way that the classical shear strain components of the Timoshenko beam theory are recovered exactly. According to the new kinematics proposed, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition. Later, a procedure for a posteriori reconstruction of all the three-dimensional stress components in the finite element analysis of thin-walled beams using the GBT is presented. The reconstruction is simple and based on the use of three-dimensional equilibrium equations and of the RCP procedure. Finally, once the stress reconstruction procedure is presented, a study of several existing issues on the constitutive relations in the GBT is carried out. Specifically, a constitutive law based on mirroring the kinematic constraints of the GBT model into a specific stress field assumption is proposed. It is shown that this method is equally valid for isotropic and orthotropic beams and coincides with the conventional GBT approach available in the literature. Later on, an analogous procedure is presented for the case of laminated beams. Lastly, as a way to improve an inherently poor description of shear deformability in the GBT, the introduction of shear correction factors is proposed. Throughout this work, numerous examples are provided to determine the validity of all the proposed contributions to the field.

Numerical loop calculations with contour deformation

Gegenstand dieser Arbeit ist die nummerische Berechnung von Schleifenintegralen welche in höheren Ordnungen der Störungstheorie auftreten.rnAnalog zur reellen Emission kann man auch in den virtuellen Beiträgen Subtraktionsterme einführen, welche die kollinearen und soften Divergenzen des Schleifenintegrals entfernen. Die Phasenraumintegration und die Schleifenintegration können dann in einer einzigen Monte Carlo Integration durchgeführt werden. In dieser Arbeit zeigen wir wie eine solche numerische Integration unter zu Hilfenahme einer Kontourdeformation durchgeführt werden kann. Ausserdem zeigen wir wie man die benötigeten Integranden mit Rekursionsformeln berechnen kann.

BCJ-relations in quantum field theory

BCJ-relations have a series of important consequences in Quantum FieldrnTheory and in Gravity. In QFT, one can use BCJ-relations to reduce thernnumber of independent colour-ordered partial amplitudes and to relate nonplanarrnand planar diagrams in loop calculations. In addition, one can usernBCJ-numerators to construct gravity scattering amplitudes through a squaringrn procedure. For these reasons, it is important to nd a prescription tornobtain BCJ-numerators without requiring a diagram by diagram approach.rnIn this thesis, after introducing some basic concepts needed for the discussion,rnI will examine the existing diagrammatic prescriptions to obtainrnBCJ-numerators. Subsequently, I will present an algorithm to construct anrneective Yang-Mills Lagrangian which automatically produces kinematic numeratorsrnsatisfying BCJ-relations. A discussion on the kinematic algebrarnfound through scattering equations will then be presented as a way to xrnnon-uniqueness problems in the algorithm.

Aspects of supergeometry in locally covariant quantum field theory

In questa tesi vengono presentati i piu recenti risultati relativi all'estensione della teoria dei campi localmente covariante a geometrie che permettano di descrivere teorie di campo supersimmetriche. In particolare, si mostra come la definizione assiomatica possa essere generalizzata, mettendo in evidenza le problematiche rilevanti e le tecniche utilizzate in letteratura per giungere ad una loro risoluzione. Dopo un'introduzione alle strutture matematiche di base, varieta Lorentziane e operatori Green-iperbolici, viene definita l'algebra delle osservabili per la teoria quantistica del campo scalare. Quindi, costruendo un funtore dalla categoria degli spazio-tempo globalmente iperbolici alla categoria delle *-algebre, lo stesso schema viene proposto per le teorie di campo bosoniche, purche definite da un operatore Green-iperbolico su uno spazio-tempo globalmente iperbolico. Si procede con lo studio delle supervarieta e alla definizione delle geometrie di background per le super teorie di campo: le strutture di super-Cartan. Associando canonicamente ad ognuna di esse uno spazio-tempo ridotto, si introduce la categoria delle strutture di super-Cartan (ghsCart) il cui spazio-tempo ridotto e globalmente iperbolico. Quindi, si mostra, in breve, come e possibile costruire un funtore da una sottocategoria di ghsCart alla categoria delle super *-algebre e si conclude presentando l'applicazione dei risultati esposti al caso delle strutture di super-Cartan in dimensione 2|2.

Use of manual adaptive remeshing in the mechanical modeling of an intraneural ganglion cyst

Intraneural Ganglion Cysts expand within in a nerve, causing neurological deficits in afflicted patients. Modeling the propagation of these cysts, originating in the articular branch and then expanding radially outward, will help prove articular theory, and ultimately allow for more purposeful treatment of this condition. In Finite Element Analysis, traditional Lagrangian meshing methods fail to model the excessive deformation that occurs in the propagation of these cysts. This report explores the method of manual adaptive remeshing as a method to allow for the use of Lagrangian meshing, while circumventing the severe mesh distortions typical of using a Lagrangian mesh with a large deformation. Manual adaptive remeshing is the process of remeshing a deformed meshed part and then reapplying loads in order to achieve a larger deformation than a single mesh can achieve without excessive distortion. The methods of manual adaptive remeshing described in this Master’s Report are sufficient in modeling large deformations.

Constraints on the effective fluid theory of stationary branes

We develop further the effective fluid theory of stationary branes. This formalism applies to stationary blackfolds as well as to other equilibrium brane systems at finite temperature. The effective theory is described by a Lagrangian containing the information about the elastic dynamics of the brane embedding as well as the hydrodynamics of the effective fluid living on the brane. The Lagrangian is corrected order-by-order in a derivative expansion, where we take into account the dipole moment of the brane which encompasses finite-thickness corrections, including transverse spin. We describe how to extract the thermodynamics from the Lagrangian and we obtain constraints on the higher-derivative terms with one and two derivatives. These constraints follow by comparing the brane thermodynamics with the conserved currents associated with background Killing vector fields. In particular, we fix uniquely the one- and two-derivative terms describing the coupling of the transverse spin to the background space-time. Finally, we apply our formalism to two blackfold examples, the black tori and charged black rings and compare the latter to a numerically generated solution.

Chiral-Scale Perturbation Theory About an Infrared Fixed Point

We review the failure of lowest order chiral SU(3)L ×SU(3)R perturbation theory χPT3 to account for amplitudes involving the f0(500) resonance and O(mK) extrapolations in momenta. We summarize our proposal to replace χPT3 with a new effective theory χPTσ based on a low-energy expansion about an infrared fixed point in 3-flavour QCD. At the fixed point, the quark condensate ⟨q̅q⟩vac ≠ 0 induces nine Nambu-Goldstone bosons: π,K,η and a QCD dilaton σ which we identify with the f0(500) resonance. We discuss the construction of the χPTσ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the ΔI = 1/2 rule in K-decays and an estimate for the Drell-Yan ratio in the infrared limit.

Introduction to Soft-Collinear Effective Theory

Among resummation techniques for perturbative QCD in the context of collider and flavor physics, soft-collinear effective theory (SCET) has emerged as both a powerful and versatile tool, having been applied to a large variety of processes, from B-meson decays to jet production at the LHC. This book provides a concise, pedagogical introduction to this technique. It discusses the expansion of Feynman diagrams around the high-energy limit, followed by the explicit construction of the effective Lagrangian - first for a scalar theory, then for QCD. The underlying concepts are illustrated with the quark vector form factor at large momentum transfer, and the formalism is applied to compute soft-gluon resummation and to perform transverse-momentum resummation for the Drell-Yan process utilizing renormalization group evolution in SCET. Finally, the infrared structure of n-point gauge-theory amplitudes is analyzed by relating them to effective-theory operators. This text is suitable for graduate students and non-specialist researchers alike as it requires only basic knowledge of perturbative QCD.

Examples of PDEs all whose points are characteristic

Let π : FM ! M be the bundle of linear frames of a manifold M. A basis Lijk , j < k, of diffeomorphism invariant Lagrangians on J1 (FM) was determined in [J. Muñoz Masqué, M. E. Rosado, Invariant variational problems on linear frame bundles, J. Phys. A35 (2002) 2013-2036]. The notion of a characteristic hypersurface for an arbitrary first-order PDE system on an ar- bitrary bred manifold π : P → M, is introduced and for the systems dened by the Euler-Lagrange equations of Lijk every hypersurface is shown to be characteristic. The Euler-Lagrange equations of the natural basis of Lagrangian densities Lijk on the bundle of linear frames of a manifold M which are invariant under diffeomorphisms, are shown to be an underdetermined PDEs systems such that every hypersurface of M is characteristic for such equations. This explains why these systems cannot be written in the Cauchy-Kowaleska form, although they are known to be formally integrable by using the tools of geometric theory of partial differential equations, see [J. Muñoz Masqué, M. E. Rosado, Integrability of the eld equations of invariant variational problems on linear frame bundles, J. Geom. Phys. 49 (2004), 119-155]

Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.