There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 - 2{epsilon} dimensions, as a Laurent expansion about {epsilon} = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/{epsilon}{sup 2} pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.

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Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

We describe the universal Monte-Carlo (parton-level) event generator WHIZARD (http://whizard.event-generator.org), version 2. The program automatically computes complete tree-level matrix elements, integrates them over phase space, evaluates distributions of observables, and generates unweighted partonic event samples. These are showered and hadronized by calling external codes, either automatically from within the program or via standard interfaces. There is no conceptual limit on the process complexity; using current hardware, the program has successfully been applied to hard scattering processes with up to eight particles in the final state. Matrix elements are computed as helicity amplitudes, so spin and color correlations are retained. For event generation, processes can be concatenated with full spin correlation, so factorized approximations to cascade decays are possible when complete matrix elements are not desired. The Standard Model, the MSSM, and many alternative models such as Little Higgs, anomalous couplings, or effects of extra dimensions or noncommutative SM extensions have been implemented. Using standard interfaces to parton shower and hadronization programs, WHIZARD covers physics at hadron, lepton, and photon colliders.

/pdf/whizard-simulating-multi-particle-processes-at-lhc-and-ilc-460o90g47h.pdf

WHIZARD—simulating multi-particle processes at LHC and ILC

We present a new tree-level matrix element generator, based on the color dressed Berends-Giele recursive relations. We discuss two new algorithms for phase space integration, dedicated to be used with large multiplicities and color sampling.

/pdf/comix-a-new-matrix-element-generator-3l413dg8pu.pdf

Comix, a new matrix element generator

Resolution of singularities for multi-loop integrals

We present the result for the finite part of the two-loop sunrise integral with unequal masses in four space-time dimensions in terms of the O(e0)-part and the O(e1)-part of the sunrise integral around two space-time dimensions. The latter two integrals are given in terms of elliptic generalisations of Clausen and Glaisher functions. Interesting aspects of the result for the O(e1)-part of the sunrise integral around two space-time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms.

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.

The two-loop sunrise graph with arbitrary masses

We present the two-loop sunrise integral with arbitrary non-zero masses in two space-time dimensions in terms of elliptic dilogarithms. We find that the structure of the result is as simple and elegant as in the equal mass case, only the arguments of the elliptic dilogarithms are modified. These arguments have a nice geometric interpretation.

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms

I consider the expansion of transcendental functions in a small parameter around rational numbers This includes in particular the expansion around half-integer values I present algorithms which are suitable for an implementation within a symbolic computer algebra system The method is an extension of the technique of nested sums The algorithms allow in addition the evaluation of binomial sums, inverse binomial sums and generalizations thereof

Stefan Weinzierl

Papers

Resolution of singularities for multi-loop integrals

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

The two-loop sunrise graph with arbitrary masses

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms

Expansion around half-integer values, binomial sums, and inverse binomial sums