13,132 research outputs found
Numerical evaluation of loop integrals
We present a new method for the numerical evaluation of arbitrary loop
integrals in dimensional regularization. We first derive Mellin-Barnes integral
representations and apply an algorithmic technique, based on the Cauchy
theorem, to extract the divergent parts in the epsilon->0 limit. We then
perform an epsilon-expansion and evaluate the integral coefficients of the
expansion numerically. The method yields stable results in physical kinematic
regions avoiding intricate analytic continuations. It can also be applied to
evaluate both scalar and tensor integrals without employing reduction methods.
We demonstrate our method with specific examples of infrared divergent
integrals with many kinematic scales, such as two-loop and three-loop box
integrals and tensor integrals of rank six for the one-loop hexagon topology
Bootstrapping pentagon functions
In PRL 116 (2016) no.6, 062001, the space of planar pentagon functions that
describes all two-loop on-shell five-particle scattering amplitudes was
introduced. In the present paper we present a natural extension of this space
to non-planar pentagon functions. This provides the basis for our pentagon
bootstrap program. We classify the relevant functions up to weight four, which
is relevant for two-loop scattering amplitudes. We constrain the first entry of
the symbol of the functions using information on branch cuts. Drawing on an
analogy from the planar case, we introduce a conjectural second-entry condition
on the symbol. We then show that the information on the function space, when
complemented with some additional insights, can be used to efficiently
bootstrap individual Feynman integrals. The extra information is read off of
Mellin-Barnes representations of the integrals, either by evaluating simple
asymptotic limits, or by taking discontinuities in the kinematic variables. We
use this method to evaluate the symbols of two non-trivial non-planar
five-particle integrals, up to and including the finite part.Comment: 24 pages + 3 pages of appendices, 2 figures, 3 tables, 4 ancillary
files, added references and corrected typos, published versio
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
We present the analytic calculation of the Master Integrals necessary to
compute the planar massive QCD corrections to Di-photon (and Di-jet) production
at hadron colliders. The masters are evaluated by means of the differential
equations method and expressed in terms of multiple polylogarithms and one- or
two-fold integrals of polylogarithms and irrational functions, up to
transcendentality four.Comment: 20 pages, ancillary file
Calculation of Massless Feynman Integrals using Harmonic Sums
A method for the evaluation of the epsilon expansion of multi-loop massless
Feynman integrals is introduced. This method is based on the Gegenbauer
polynomial technique and the expansion of the Gamma function in terms of
harmonic sums. Algorithms for the evaluation of nested and harmonic sums are
used to reduce the expressions to get analytical or numerical results for the
expansion coefficients. Methods to increase the precision of numerical results
are discussed.Comment: 30 pages, 6 figures; Minor typos corrected, references added.
Published in Computer Physics Communication
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