172 research outputs found
Global Poles of the Two-Loop Six-Point N=4 SYM integrand
Recently, a recursion relation has been developed, generating the
four-dimensional integrand of the amplitudes of N=4 supersymmetric Yang-Mills
theory for any number of loops and legs. In this paper, I provide a comparison
of the prediction for the two-loop six-point maximally helicity-violating (MHV)
integrand against the result obtained by use of the leading singularity method.
The comparison is performed numerically for a large number of randomly selected
momenta and in all cases finds agreement between the two results to high
numerical accuracy.Comment: 32+34 pages, 16 figures, 1 notebook; minor typos corrected, ref.
added; version accepted by Phys. Rev.
Integration-by-parts reductions from unitarity cuts and algebraic geometry
We show that the integration-by-parts reductions of various two-loop integral
topologies can be efficiently obtained by applying unitarity cuts to a specific
set of subgraphs and solving associated polynomial (syzygy) equations.Comment: 5 pages, 1 figure; a Mathematica package implementing the algorithm
is attached as an ancillary file; v3: minor change
MultivariateResidues - a Mathematica package for computing multivariate residues
We present the Mathematica package MultivariateResidues, which allows for the
efficient evaluation of multivariate residues based on methods from
computational algebraic geometry. Multivariate residues appear in several
contexts of scattering amplitude computations. Examples include applications to
the extraction of master integral coefficients from maximal unitarity cuts, the
construction of canonical bases of loop integrals and the construction of tree
amplitudes from scattering equations.Comment: 7 pages, 2 figures, contribution to the proceedings of the 13th
International Symposium on Radiative Corrections (RADCOR 2017
Position-space cuts for Wilson line correlators
We further develop the formalism for taking position-space cuts of eikonal
diagrams introduced in [Phys.Rev.Lett. 114 (2015), no. 18 181602,
arXiv:1410.5681]. These cuts are applied directly to the position-space
representation of any such diagram and compute its discontinuity to the leading
order in the dimensional regulator. We provide algorithms for computing the
position-space cuts and apply them to several two- and three-loop eikonal
diagrams, finding agreement with results previously obtained in the literature.
We discuss a non-trivial interplay between the cutting prescription and
non-Abelian exponentiation. We furthermore discuss the relation of the
imaginary part of the cusp anomalous dimension to the static interquark
potential.Comment: 39+18 pages, 16 figures; elaborated the discussion of the comparison
of numerical and analytic results for the three-gluon vertex diagram in the
caption of fig. 16; version to be published in JHE
Imaginary parts and discontinuities of Wilson line correlators
We introduce a notion of position-space cuts of eikonal diagrams, the set of
diagrams appearing in the perturbative expansion of the correlator of a set of
straight semi-infinite Wilson lines. The cuts are applied directly to the
position-space representation of any such diagram and compute its imaginary
part to the leading order in the dimensional regulator. Our cutting
prescription thus defines a position-space analog of the standard
momentum-space Cutkosky rules. Unlike momentum-space cuts which put internal
lines on shell, position-space cuts constrain a number of the gauge bosons
exchanged between the energetic partons to be lightlike, leading to a vanishing
and a non-vanishing imaginary part for space- and timelike kinematics,
respectively.Comment: 5 pages, 2 figures; minor changes; version published in PR
Cristal and Azurite: new tools for integration-by-parts reductions
Scattering amplitudes computed at a fixed loop order, along with any other
object computed in perturbative quantum field theory, can be expressed as a
linear combination of a finite basis of loop integrals. To compute loop
amplitudes in practice, such a basis of integrals must be determined. We
discuss Azurite (A ZURich-bred method for finding master InTEgrals), a publicly
available package for finding bases of loop integrals. We also discuss Cristal
(Complete Reduction of IntegralS Through All Loops), a future package that
produces the complete integration-by-parts reductions.Comment: 7 pages, 3 figures. Contribution to the proceedings of RADCOR 2017,
25-29 September 2017, St. Gilgen, Austri
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
Differential equations for loop integrals in Baikov representation
We present a proof that differential equations for Feynman loop integrals can
always be derived in Baikov representation without involving dimension-shift
identities. We moreover show that in a large class of two- and three-loop
diagrams it is possible to avoid squared propagators in the intermediate steps
of setting up the differential equations.Comment: 11 pages, two-column format, 4 figures. Minor changes; journal
versio
Two-Loop Maximal Unitarity with External Masses
We extend the maximal unitarity method at two loops to double-box basis
integrals with up to three external massive legs. We use consistency equations
based on the requirement that integrals of total derivatives vanish. We obtain
unique formulae for the coefficients of the master double-box integrals. These
formulae can be used either analytically or numerically.Comment: 41 pages, 7 figures; small corrections, final journal versio
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