Aspects of high energy scattering

Scattering amplitudes in quantum field theories are of widespread interest, due to a large number of theoretical and phenomenological applications. Much is known about the possible behaviour of amplitudes, that is independent of the details of the underlying theory. This knowledge is often neglected in modern QFT courses, and the aim of these notes - aimed at graduate students - is to redress this. We review the possible singularities that amplitudes can have, before examining the generic behaviour that can arise in the high-energy limit. Finally, we illustrate the results using examples from QCD and gravity.

DIFFERENTIAL-ALGEBRAIC JET SPACES PRESERVE INTERNALITY TO THE CONSTANTS

Suppose p is the generic type of a differential-algebraic jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constants. This strengthening, which was originally called “being Moishezon to the constants” in [9] but is here renamed preserving internality to the constants, is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. An example is given showing that only a generic analogue holds in the differential-algebraic case: there is a finite dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

Product dynamics for homoclinic attractors

Heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system. Even for cycles between equilibria, it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle. For more–complicated cycles between chaotic sets, the non–trivial dynamics of the ‘nodes’ can interact with that of the ‘connections’. This paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics. Using a precise analytic description of a general planar homoclinic attractor, we are able to obtain a number of results for direct product systems. We show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor, the product of the attractors is a minimal Milnor attractor for the product. On the other hand, we present evidence to show that for the product of two homoclinic attractors, typically only a small subset of the product of the attractors is an attractor for the product system.

Pascal's triangles in Abelian and hyperbolic groups

Given a group G and a finite generating set G, we take pG: G → Z to be the function which counts the number of geodesics for each group element g. This generalizes Pascal's triangle. We compute pG for word hyperbolic and describe generic behaviour in abelian groups.

Microstructure suspended in three-dimensional flows

The dynamical behaviour of stretchable, orientable microstructure suspended in a general three-dimensional fluid flow is investigated. Model equations given by Olbricht, Rallison & Leal (1982) are examined in the case of microstructure travelling through arbitrarily complicated flows of the carrier fluid. As in the two-dimensional analysis of Szeri, Wiggins & Leal (1991), one must first treat the orientation dynamics problem; only then can the equation for stretch of the microstructure be analyzed rationally. In three-dimensional flows that are steady in the Lagrangian frame, attractors for the orientation dynamics are shown to be equilibria or limit cycles; this asymptotic behaviour was first deduced by Bretherton (1962). In three-dimensional flows that are time periodic in the Lagrangian frame (e.g. recirculating flows), the orientation dynamics may be characterized by periodic or quasi-periodic attractors. Thus, robust (generic) behaviour in these cases is always characterized by a single global attractor; there is no asymptotic dependence of orientation dynamics on the initial orientation. The type of asymptotic orientation dynamics – steady, periodic, or quasi-periodic - is signified by a simple criterion. Details of the relevant bifurcations, as well as history-dependent strong flow criteria are developed. Examples which illustrate the various types of behaviour are given.

Helicity and the Călugăreanu invariant

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.