The Structure of Amplitudes in N=4 SYM

The study of amplitudes and related quantities in the N = 4 Super Yang-Mills theory is a subject undergoing rapid evolution at the moment. In this work we present a review of some of the key ideas and concepts which we use to calculate `l-loop, n-point amplitudes of varying helicity. We show that performing a restriction on the external data of being in 1 + 1-dimensions allows remarkably compact expressions to be obtained at both MHV and NMHV levels. We use this data to motivate in 1 + 1-dimensions remarkably simple formulae for all collinear-limits and ultimately a universal uplifting formula which generates all n-point amplitudes of a particular loop-order and helicity configuration from a small set of lower-loop amplitudes. We also use the mechanism of the correlation function ↔ amplitude duality to construct the integrand for the five-point amplitude in full four-dimensional kinematics to six-loops in the parity-even sector and five-loops in the parity-odd sector. Finally we consider a rewriting of certain known momentum-twistor amplitudes in terms of bi-twistor, six-dimensional X-variables and dimensionally regularise these equations to match known O(ε) results. From this we make some observations about the requirements for this process to be successful in the limited number of cases where the full O(ε) solution is known and provide an ansatz for constructing the terms for more complicated amplitudes

Local integrands for the five-point amplitude in planar N=4 SYM up to five loops

Integrands for colour ordered scattering amplitudes in planar N=4 SYM are dual to those of correlation functions of the energy-momentum multiplet of the theory. The construction can relate amplitudes with different numbers of legs. By graph theory methods the integrand of the four-point function of energy-momentum multiplets has been constructed up to six loops in previous work. In this article we extend this analysis to seven loops and use it to construct the full integrand of the five-point amplitude up to five loops, and in the parity even sector to six loops. All results, both parity even and parity odd, are obtained in a concise local form in dual momentum space and can be displayed efficiently through graphs. We have verified agreement with other local formulae both in terms of supertwistors and scalar momentum integrals as well as BCJ forms where those exist in the literature, i.e. up to three loops. Finally we note that the four-point correlation function can be extracted directly from the four-point amplitude and so this uncovers a direct link from four- to five-point amplitudes.Comment: 29 pages LaTeX, 8 figure

A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions

The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using an example

Modelling inelastic granular media using Dynamical Density Functional Theory

We construct a new mesoscopic model for granular media using Dynamical Density Functional Theory (DDFT). The model includes both a collision operator to incorporate inelasticity and the Helmholtz free energy functional to account for external potentials, interparticle interactions and volume exclusion. We use statistical data from event-driven microscopic simulations to determine the parameters not given analytically by the closure relations used to derive the DDFT. We numerically demonstrate the crucial effects of each term in the DDFT, and the importance of including an accurately parametrised pair correlation function

Experimental realization of catalytic CH_4 hydroxylation predicted for an iridium NNC pincer complex, demonstrating thermal, protic, and oxidant stability

A discrete, air, protic, and thermally stable (NNC)Ir(III) pincer complex was synthesized that catalytically activates the CH bond of methane in trifluoroacetic acid; functionalization using NaIO_4 and KIO_3 gives the oxy-ester

Synthesis and Characterization of the k^2-acac-O,O Complex Os_(IV)(acac)_2PhCl and Study of CH Activation with Benzene

We have synthesized and fully characterized the air-stable complex (κ^2-acac-O,O)2Os^(IV)(Ph)Cl (Cl-1-Ph; acac-O,O = acetylacetonate), which reacts with C_6D_6 to generate Cl-1-Ph-d_5 in high yield and catalyzes the H/D exchange reaction between benzene and toluene-d_8 upon heating to 140 °C. To our knowledge, this is the first example of stoichiometric and catalytic, homogeneous, intermolecular CH activation of arenes by a discrete Os complex. The reactions show extended induction periods. DFT studies of Cl-1-Ph and cis-(κ^2-acac-O,O)_2Os^(III)(C_6H_5)(C_6D_6) (cis-(C_6D_6)-2-Ph) found a mechanism involving CH activation by traces of Os(III) and Cl atom transfer between Cl-1-Ph and cis-(C_6D_6)-2-Ph. Experimental data showing that addition of reductants eliminates the induction periods suggest that CH activation occurs from an oxidation state lower than Os^(IV), consistent with the DFT predictions. Consistent with a Cl atom transfer mechanism, the triflate analogue of Cl-1-Ph, OTf-1-Ph, does not undergo a stoichiometric or catalytic reaction with C_6D_6

On the weak coupling spectrum of N=2 supersymmetric SU(n) gauge theory

The weak coupling spectrum of BPS saturated states of pure $N=2$ supersymmetric SU$(n)$ gauge theory is investigated. The method uses known results on the dyon spectrum of the analogous theory with $N=4$ supersymmetry, along with the action on these states of the semi-classical monodromy transformations. For dyons whose magnetic charge is not a simple root of the Lie algebra, it is found that the weak coupling region is divided into a series of domains, for which the dyons have different electric charge, separated by walls on which the dyons decay. The proposed spectrum is shown to be consistent with the exact solution of the theory at strong coupling in the sense that the states at weak coupling can account for the singularities at strong coupling.Comment: 19 pages, plain tex with macro included, 6 figures included using psfig.te

Two Time Scales for The Price Of One (Almost)

Although differences exist between seasonal- and decadal-scale climate variability, predictability, and prediction, investment in observations, prediction systems, and decision systems for either time scale can benefit both

Semi-classical Quantization in N=4 Supersymmetric Yang-Mills Theory and Duality

At a generic point in the moduli space of vacua of an N=4 supersymmetric gauge theory with arbitrary gauge group the Higgs force does not cancel the magneto-static force between magnetic monopoles of distinct charge. As a consequence the moduli space of magnetically charged solutions is related in a simple way to those of the SU(2) theory. This leads to a rather simple test of S-duality. On certain subspaces of the moduli space of vacua the forces between distinct monopoles cancel and the test of S-duality becomes more complicated.Comment: 11 pages, plain tex with macro included, one Postscript figure, uses psfig.te