Differential equations from unitarity cuts: nonplanar hexa-box integrals

We compute $\epsilon$-factorized differential equations for all dimensionally-regularized integrals of the nonplanar hexa-box topology, which contribute for instance to 2-loop 5-point QCD amplitudes. A full set of pure integrals is presented. For 5-point planar topologies, Gram determinants which vanish in $4$ dimensions are used to build compact expressions for pure integrals. Using unitarity cuts and computational algebraic geometry, we obtain a compact IBP system which can be solved in 8 hours on a single CPU core, overcoming a major bottleneck for deriving the differential equations. Alternatively, assuming prior knowledge of the alphabet of the nonplanar hexa-box, we reconstruct analytic differential equations from 30 numerical phase-space points, making the computation almost trivial with current techniques. We solve the differential equations to obtain the values of the master integrals at the symbol level. Full results for the differential equations and solutions are included as supplementary material.Comment: 31 pages, 2 figures. Version 2: final journal version; includes solutions to differential equation

Analytical modelling in Dynamo

BIM is applied as modern database for civil engineering. Its recent development allows to preserve both structure geometrical and analytical information. The analytical model described in the paper is derived directly from BIM model of a structure automatically but in most cases it requires manual improvements before being sent to FEM software. Dynamo visual programming language was used to handle the analytical data. Authors developed a program which corrects faulty analytical model obtained from BIM geometry, thus providing better automation for preparing FEM model. Program logic is explained and test cases shown

Comparing efficient computation methods for massless QCD tree amplitudes: Closed Analytic Formulae versus Berends-Giele Recursion

Recent advances in our understanding of tree-level QCD amplitudes in the massless limit exploiting an effective (maximal) supersymmetry have led to the complete analytic construction of tree-amplitudes with up to four external quark-anti-quark pairs. In this work we compare the numerical efficiency of evaluating these closed analytic formulae to a numerically efficient implementation of the Berends-Giele recursion. We compare calculation times for tree-amplitudes with parton numbers ranging from 4 to 25 with no, one, two and three external quark lines. We find that the exact results are generally faster in the case of MHV and NMHV amplitudes. Starting with the NNMHV amplitudes the Berends-Giele recursion becomes more efficient. In addition to the runtime we also compared the numerical accuracy. The analytic formulae are on average more accurate than the off-shell recursion relations though both are well suited for complicated phenomenological applications. In both cases we observe a reduction in the average accuracy when phase space configurations close to singular regions are evaluated. We believe that the above findings provide valuable information to select the right method for phenomenological applications.Comment: 22 pages, 9 figures, Mathematica package GGT.m and example notebook is included in submissio

Solving Recurrence Relations for Multi-Loop Feynman Integrals

We study the problem of solving integration-by-parts recurrence relations for a given class of Feynman integrals which is characterized by an arbitrary polynomial in the numerator and arbitrary integer powers of propagators, {\it i.e.}, the problem of expressing any Feynman integral from this class as a linear combination of master integrals. We show how the parametric representation invented by Baikov can be used to characterize the master integrals and to construct an algorithm for evaluating the corresponding coefficient functions. To illustrate this procedure we use simple one-loop examples as well as the class of diagrams appearing in the calculation of the two-loop heavy quark potential.Comment: 24 pages, 5 ps figures, references added, minor modifications, published versio

Combinatorial Applications of the Subspace Theorem

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and the construction of transcendental numbers. But its usefulness extends beyond the realms of number theory. Other applications of the Subspace Theorem include linear recurrence sequences and finite automata. In fact, these structures are closely related to each other and the construction of transcendental numbers. The Subspace Theorem also has a number of remarkable combinatorial applications. The purpose of this paper is to give a survey of some of these applications including sum-product estimates and bounds on unit distances. The presentation will be from the point of view of a discrete mathematician. We will state a number of variants of the Subspace Theorem below but we will not prove any of them as the proofs are beyond the scope of this work. However we will give a proof of a simplified special case of the Subspace Theorem which is still very useful for many problems in discrete mathematics