50 research outputs found
The classification of two-loop integrand basis in pure four-dimension
In this paper, we have made the attempt to classify the integrand basis of
all two-loop diagrams in pure four-dimension space-time. Our classification
includes the topology of two-loop diagrams which determines the structure of
denominators, and the set of numerators under different kinematic
configurations of external momenta by using Gr\"{o}bner basis method. In our
study, the variety defined by setting all propagators to on-shell has played an
important role. We discuss the structure of variety and how it splits to
various irreducible branches when external momenta at each corner of diagrams
satisfy some special kinematic conditions. This information is crucial to the
numerical or analytical fitting of coefficients for integrand basis in
reduction process.Comment: 52 pages, 9 figures. v2 reference added, v3 published versio
Form Factor and Boundary Contribution of Amplitude
The boundary contribution of an amplitude in the BCFW recursion relation can
be considered as a form factor involving boundary operator and unshifted
particles. At the tree-level, we show that by suitable construction of
Lagrangian, one can relate the leading order term of boundary operators to some
composite operators of N=4 super-Yang-Mills theory, then the computation of
form factors is translated to the computation of amplitudes. We compute the
form factors of these composite operators through the computation of
corresponding double trace amplitudes.Comment: 38 pages, 6 figure
Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
For a physical field theory, the tree-level amplitudes should possess only
single poles. However, when computing amplitudes with Cachazo-He-Yuan (CHY)
formulation, individual terms in the intermediate steps will contribute
higher-order poles. In this paper, we investigate the cancelation of
higher-order poles in CHY formula with Pfaffian as the building block. We
develop a diagrammatic rule for expanding the reduced Pfaffian. Then by
organizing diagrams in appropriate groups and applying the cross-ratio
identities, we show that all potential contributions to higher-order poles in
the reduced Pfaffian are canceled out, i.e., only single poles survive in
Yang-Mills theory and gravity. Furthermore, we show the cancelations of
higher-order poles in other field theories by introducing appropriate
truncations, based on the single pole structure of Pfaffian.Comment: 30 pages,6 figures,1 table, footnote adde
Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
In this paper, we generalize the unitarity method to two-loop diagrams and
use it to discuss the integral bases of reduction. To test out method, we focus
on the four-point double-box diagram as well as its related daughter diagrams,
i.e., the double-triangle diagram and the triangle-box diagram. For later two
kinds of diagrams, we have given complete analytical results in general
(4-2\eps)-dimension.Comment: 52 pages, 1 figur
Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
This paper studies the global structure of algebraic curves defined by
generalized unitarity cut of four-dimensional three-loop diagrams with eleven
propagators. The global structure is a topological invariant that is
characterized by the geometric genus of the algebraic curve. We use the
Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with
the help of techniques involving convex hull polytopes and numerical algebraic
geometry. Some interesting properties of genus for arbitrary loop orders are
also explored where computing the genus serves as an initial step for integral
or integrand reduction of three-loop amplitudes via an algebraic geometric
approach.Comment: 35pages, 10 figures, version appeared in JHE
Expansion of Einstein-Yang-Mills Amplitude
In this paper, we provide a thorough study on the expansion of single trace
Einstein-Yang-Mills amplitudes into linear combination of color-ordered
Yang-Mills amplitudes, from various different perspectives. Using the gauge
invariance principle, we propose a recursive construction, where EYM amplitude
with any number of gravitons could be expanded into EYM amplitudes with less
number of gravitons. Through this construction, we can write down the complete
expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes.
As a byproduct, we are able to write down the polynomial form of BCJ numerator,
i.e., numerators satisfying the color-kinematic duality, for Yang-Mills
amplitude. After the discussion of gauge invariance, we move to the BCFW
on-shell recursion relation and discuss how the expansion can be understood
from the on-shell picture. Finally, we show how to interpret the expansion from
the aspect of KLT relation and the way of evaluating the expansion coefficients
efficiently.Comment: 50 pages, 1 figure, Revised versio