Azurite: An algebraic geometry based package for finding bases of loop integrals

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package {\sc Azurite} ({\bf A ZUR}ich-bred method for finding master {\bf I}n{\bf TE}grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems {\sc Singular} and {\sc Mathematica}. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts.Comment: Version 1.1.0 of the package Azurite, with parallel computations. It can be downloaded from https://bitbucket.org/yzhphy/azurite/raw/master/release/Azurite_1.1.0.tar.g

SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop

SecDec is a program which can be used for the factorization of dimensionally regulated poles from parametric integrals, in particular multi-loop integrals, and the subsequent numerical evaluation of the finite coefficients. Here we present version 3.0 of the program, which has major improvements compared to version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability.Comment: 46 pages, version to appear in Comput.Phys.Com

On the Numerical Evaluation of Loop Integrals With Mellin-Barnes Representations

An improved method is presented for the numerical evaluation of multi-loop integrals in dimensional regularization. The technique is based on Mellin-Barnes representations, which have been used earlier to develop algorithms for the extraction of ultraviolet and infrared divergencies. The coefficients of these singularities and the non-singular part can be integrated numerically. However, the numerical integration often does not converge for diagrams with massive propagators and physical branch cuts. In this work, several steps are proposed which substantially improve the behavior of the numerical integrals. The efficacy of the method is demonstrated by calculating several two-loop examples, some of which have not been known before.Comment: 13 pp. LaTe

Numerical Loop-Tree Duality: contour deformation and subtraction

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.Comment: 87 page

Maximal Cuts in Arbitrary Dimension

We develop a systematic procedure for computing maximal unitarity cuts of multiloop Feynman integrals in arbitrary dimension. Our approach is based on the Baikov representation in which the structure of the cuts is particularly simple. We examine several planar and nonplanar integral topologies and demonstrate that the maximal cut inherits IBPs and dimension shift identities satisfied by the uncut integral. Furthermore, for the examples we calculated, we find that the maximal cut functions from different allowed regions, form the Wronskian matrix of the differential equations on the maximal cut.Comment: typos corrected, more references adde

Automatic Computation of Feynman Diagrams

Quantum corrections significantly influence the quantities observed in modern particle physics. The corresponding theoretical computations are usually quite lengthy which makes their automation mandatory. This review reports on the current status of automatic calculation of Feynman diagrams in particle physics. The most important theoretical techniques are introduced and their usefulness is demonstrated with the help of simple examples. A survey over frequently used programs and packages is provided, discussing their abilities and fields of applications. Subsequently, some powerful packages which have already been applied to important physical problems are described in more detail. The review closes with the discussion of a few typical applications for the automated computation of Feynman diagrams, addressing current physical questions like properties of the $Z$ and Higgs boson, four-loop corrections to renormalization group functions and two-loop electroweak corrections.Comment: Latex, 62 pages. Typos corrected, references updated and some comments added. Vertical offset changed. The complete paper is also available via anonymous ftp at ftp://ttpux2.physik.uni-karlsruhe.de/ttp98/ttp98-41/ or via www at http://www-ttp.physik.uni-karlsruhe.de/Preprints

LUSIFER: a LUcid approach to SIx-FERmion production

LUSIFER is a Monte Carlo event generator for all processes e+e-->6fermions, which is based on the multi-channel Monte Carlo integration technique and employs the full set of tree-level diagrams. External fermions are taken to be massless, but can be arbitrarily polarized. The calculation of the helicity amplitudes and of the squared matrix elements is presented in a compact way. Initial-state radiation is included at the leading logarithmic level using the structure-function approach. The discussion of numerical results contains a comprehensive list of cross sections relevant for a 500GeV collider, including a tuned comparison to results obtained with the combination of the WHIZARD and MADGRAPH packages as far as possible. Moreover, for off-shell top-quark pair production and the production of a Higgs boson in the intermediate mass range we additionally discuss some phenomenologically interesting distributions. Finally, we numerically analyze the effects of gauge-invariance violation by comparing various ways of introducing decay widths of intermediate top quarks, gauge and Higgs bosons.Comment: 39 pages, latex, 14 postscript files, some minor misprints corrected, version to appear in Nucl.Phys.

Reduction and evaluation of two-loop graphs with arbitrary masses

We describe a general analytic-numerical reduction scheme for evaluating any 2-loop diagrams with general kinematics and general renormalizable interactions, whereby ten special functions form a complete set after tensor reduction. We discuss the symmetrical analytic structure of these special functions in their integral representation, which allows for optimized numerical integration. The process Z -> bb is used for illustration, for which we evaluate all the 3-point, non-factorizable g^2*alpha_s mixed electroweak-QCD graphs, which depend on the top quark mass. The isolation of infrared singularities is detailed, and numerical results are given for all two-loop three-point graphs involved in this process