18,884 research outputs found
Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
The search for a theory of the S-Matrix has revealed surprising geometric
structures underlying amplitudes ranging from the worldsheet to the
amplituhedron, but these are all geometries in auxiliary spaces as opposed to
kinematic space where amplitudes live. In this paper, we propose a novel
geometric understanding of amplitudes for a large class of theories. The key is
to think of amplitudes as differential forms directly on kinematic space. We
explore this picture for a wide range of massless theories in general spacetime
dimensions. For the bi-adjoint cubic scalar, we establish a direct connection
between its "scattering form" and a classic polytope--the associahedron--known
to mathematicians since the 1960's. We find an associahedron living naturally
in kinematic space, and the tree amplitude is simply the "canonical form"
associated with this "positive geometry". Basic physical properties such as
locality, unitarity and novel "soft" limits are fully determined by the
geometry. Furthermore, the moduli space for the open string worldsheet has also
long been recognized as an associahedron. We show that the scattering equations
act as a diffeomorphism between this old "worldsheet associahedron" and the new
"kinematic associahedron", providing a geometric interpretation and novel
derivation of the bi-adjoint CHY formula. We also find "scattering forms" on
kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual
to the color-dressed amplitudes despite having no explicit color factors. This
is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic
wedge products in the scattering forms satisfy the same Jacobi relations as
color factors. Finally, our scattering forms are well-defined on the
projectivized kinematic space, a property that provides a geometric origin for
color-kinematics duality.Comment: 77 pages, 25 figures; v2, corrected discussion of worldsheet
associahedron canonical for
The amplituhedron from momentum twistor diagrams
We propose a new diagrammatic formulation of the all-loop scattering
amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor
diagrams". These are on-shell-diagrams obtained by gluing trivalent black and
white vertices defined in momentum twistor space, which, in the reduced diagram
case, are known to be related to diagrams in the original twistor space. The
new diagrams are manifestly Yangian invariant, and they naturally represent
factorization and forward-limit contributions in the all-loop BCFW recursion
relations in momentum twistor space, in a fashion that is completely different
from those in momentum space. We show how to construct and evaluate
momentum-twistor diagrams, and how to use them to obtain tree-level amplitudes
and loop-level integrands; in particular for the latter we identify an isolated
bubble-structure for each loop variable, arising from a forward limit, or
entangled removal of particles. From a given diagram one can directly read off
the C, D matrices via a generalized "boundary measurement"; this in turn
determines a cell in the amplituhedron associated with the amplitude, and our
diagrammatic representations of the amplitude can provide triangulations of the
amplituhedron with generally very intricate geometries. To demonstrate the
computational power of the formalism, we give explicit results for general
two-loop integrands, and the cells of the complete amplituhedron for two-loop
MHV amplitudes.Comment: 39 pages, 34 figure
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