Antigravity Agent
1881 lines
89 KiB
TeX
1881 lines
89 KiB
TeX
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\title{\hspace{-0.0cm}{\LARGE The Amplituhedron}}
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\author{\vspace{-.5cm}Nima Arkani-Hamed$^{a}$ and Jaroslav Trnka$^{b}$\\
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{\footnotesize{\it $^{a}$ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA}\\
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{\it $^{b}$ California Institute of Technology, Pasadena, CA 91125,
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USA}}\vspace{-.5cm}} \preprint{2013}
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\abstract{Perturbative scattering amplitudes in gauge theories have
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remarkable simplicity and hidden infinite dimensional symmetries
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that are completely obscured in the conventional formulation of
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field theory using Feynman diagrams. This suggests the existence of
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a new understanding for scattering amplitudes where
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locality and unitarity do not play a central role but are derived
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consequences from a different starting point. In this note we
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provide such an understanding for ${\cal N} = 4$ SYM scattering
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amplitudes in the planar limit, which we identify as ``the volume"
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of a new mathematical object--the Amplituhedron--generalizing the
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positive Grassmannian. Locality and unitarity emerge hand-in-hand
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from positive geometry.}
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\preprint{CALT-68-2872}
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\begin{document}
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\newpage
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\section{Scattering Without Space-Time}
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Scattering amplitudes in gauge theories are amongst the most
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fundamental observables in physics. The textbook approach to
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computing these amplitudes in perturbation theory, using Feynman
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diagrams, makes locality and unitarity as manifest as possible, at
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the expense of introducing large amounts of gauge redundancy into
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our description of the physics, leading to an explosion of apparent
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complexity for the computation of amplitudes for all but the very
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simplest processes. Over the last quarter-century it has become
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clear that this complexity is a defect of the Feynman diagram
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approach to this physics, and is not present in the final amplitudes
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themselves, which are astonishingly simpler than indicated from the
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diagrammatic expansion \cite{PT,Z1,Z2,Witten:2003nn, CSW,
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BCFW1,BCFW2}.
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This has been best understood for maximally supersymmetric gauge
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theories in the planar limit. Planar ${\cal N} = 4$ SYM has been
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used as a toy model for real physics in many guises, but as toy
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models go, its application to scattering amplitudes is closer to the
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real world than any other. For instance the leading tree
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approximation to scattering amplitudes is identical to ordinary
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gluon scattering, and the most complicated part of loop amplitudes,
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involving virtual gluons, is also the same in ${\cal N} = 4$ SYM as
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in the real world.
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Planar ${\cal N} = 4$ SYM amplitudes turn out to be especially
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simple and beautiful, enjoying the hidden symmetry of dual superconformal
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invariance\cite{DCI1,DCI2}, associated with a dual interpretation of
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scattering amplitudes as a supersymmetric Wilson loop
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\cite{WL1,WL2,Alday:2010zy}. Dual superconformal symmetry combines
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with the ordinary conformal symmetry to generate an infinite
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dimension ``Yangian" symmetry \cite{Yangian}. Feynman diagrams
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conceal this marvelous structure precisely as a consequence of
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making locality and unitarity manifest. For instance, the Yangian
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symmetry is obscured in either one of the standard physical
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descriptions either as a``scattering amplitude" in one space-time or
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a ``Wilson-loop" in its dual.
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This suggests that there must be a different formulation of the
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physics, where locality and unitarity do not play a central role,
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but emerge as derived features from a different starting point. A
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program to find a reformulation along these lines was initiated in
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\cite{N1,N2}, and in the context of a planar ${\cal N} = 4$ SYM was
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pursued in \cite{N3,N4,N5}, leading to a new physical and
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mathematical understanding of scattering amplitudes \cite{N6}. This
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picture builds on BCFW recursion relations for tree \cite{BCFW1,
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BCFW2} and loop \cite{N5,Rutger} amplitudes, and represents the
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amplitude as a sum over basic building blocks, which can be
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physically described as arising from gluing together the elementary
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three-particle amplitudes to build more complicated on-shell
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processes. These ``on-shell diagrams" (which are essentially the
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same as the ``twistor diagrams" of \cite{TD1,TD2,N3}) are remarkably
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connected with ``cells" of a beautiful new structure in algebraic
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geometry, that has been studied by mathematicians over the past
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number of years, known as the positive Grassmannian \cite{alex, N6}.
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The on-shell building blocks can not be associated with local
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space-time processes. Instead, they enjoy all the symmetries of the
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theory, as made manifest by their connection with the
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Grassmannian--indeed, the infinite dimensional Yangian symmetry is
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easily seen to follow from ``positive" diffeomorphisms \cite{N6}.
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While these developments give a complete understanding for the
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on-shell building blocks of the amplitude, they do not go further to
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explain {\it why} the building blocks have to be combined in a
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particular way to determine the full amplitude itself. Indeed, the
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particular combination of on-shell diagrams is dictated by {\it
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imposing} that the final result is local and unitary--locality and
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unitarity specify the singularity structure of the amplitude, and
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this information is {\it used} to determine the full integrand. This
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is unsatisfying, since we want to see locality and unitarity emerge
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from more primitive ideas, not merely use them to obtain the
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amplitude.
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An important clue \cite{N4,A1,N6} pointing towards a deeper understanding is that
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the on-shell representation of scattering
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amplitudes is not unique: the recursion relations can be solved in
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many different ways, and so the final amplitude can be expressed as
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a sum of on-shell processes in different ways as well. The on-shell
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diagrams satisfy remarkable identities--now most easily understood
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from their association with cells of the positive Grassmannian--that
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can be used to establish these equivalences. This observation led
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Hodges \cite{A1} to a remarkable observation for the simplest case of
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``NMHV" tree amplitudes, further developed in \cite{N7}: the amplitude
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can be thought of as the volume of a certain polytope in momentum
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twistor space. However there was no a priori
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understanding of the origin of this polytope, and the picture
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resisted a direct generalization to more general trees or to loop amplitudes.
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Nonetheless, the polytope idea motivated a continuing search for a geometric representation of the amplitude
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as ``the volume" of ``some canonical region" in
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``some space", somehow related to the positive Grassmannian, with
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different ``triangulations" of the space corresponding to different
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natural decompositions of the amplitude into building blocks.
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In this note we finally realize this picture. We will introduce a
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new mathematical object whose ``volume" directly computes the
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scattering amplitude. We call this object the ``Amplituhedron", to
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denote its connection both to scattering amplitudes and positive
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geometry. The amplituhedron can be given a self-contained definition
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in a few lines as done below in section 9. We will motivate its
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definition from elementary considerations, and show how scattering
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amplitudes are extracted from it.
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Everything flows from generalizing the notion of the ``inside of a
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triangle in a plane". The first obvious generalization is
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to the inside of a simplex in projective space, which further
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extends to the positive Grassmannian. The second generalization
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is to move from triangles to convex polygons, and then extend this
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into the Grassmannian. This gives us the amplituhedron for tree
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amplitudes, generalizing the positive Grassmannian by extending the
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notion of positivity to include external kinematical data. The full
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amplituhedron at all loop order further generalizes the notion of
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positivity in a way motivated by the natural idea of ``hiding
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particles".
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Another familiar notion associated with triangles and polygons is
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their area. This is more naturally described in a projective way by
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a canonical 2-form with logarithmic singularities on the boundaries
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of the polygon. This form also generalizes to the full
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amplituhedron, and determines the (integrand of) the scattering
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amplitude. The geometry of the amplituhedron is completely bosonic,
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so the extraction of the superamplitude from this canonical form
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involves a novel treatment of supersymmetry, directly motivated by
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the Grassmannian structure.
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The connection between the amplituhedron and scattering amplitudes
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is a conjecture which has passed a large number of non-trivial
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checks, including an understanding of how locality and unitarity
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arise as consequences of positivity. Our purpose in this note is to
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motivate and give the complete definition of the amplituhedron and
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its connection to the superamplitude in planar ${\cal N}=4$ SYM. The
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discussion will be otherwise telegraphic and few details or examples
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will be given. In two accompanying notes \cite{Into, Threeviews}, we
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will initiate a systematic exploration of various aspects of the
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associated geometry and physics. A much more thorough exposition of
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these ideas, together with many examples worked out in detail, will
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be presented in \cite{Long}.
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\subsection*{Notation}
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The external data for massless $n$ particle scattering amplitudes
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(for an excellent review see \cite{review}) are labeled as
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$|\lambda_a,\tilde \lambda_a, \tilde \eta_a \rangle$ for $a=1,
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\dots, n$. Here $\lambda_a, \tilde \lambda_a$ are the
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spinor-helicity variables, determining null momenta $p_a^{A \dot{A}}
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= \lambda_a^A \tilde \lambda_a^{\dot{A}}$. The $\tilde \eta_a$ are
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(four) grassmann variables for on-shell superspace. The component
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of the color-stripped superamplitude with weight $4(k+2)$ in the $\tilde \eta$'s is
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$M_{n,k}$. We can write \be M_{n,k}(\lambda_a, \tilde \lambda_a,
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\tilde \eta_a) = \frac{\delta^4(\sum_a \lambda_a \tilde \lambda_a)
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\delta^8(\sum_a \lambda_a \tilde \eta_a)}{\langle 1 2 \rangle \dots
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\langle n 1 \rangle} \times {\cal M}_{n,k}( z_a, \eta_a) \ee where
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$(z_a, \eta_a)$ are the (super) ``momentum-twistor" variables
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\cite{A1}, with $ z_a = \left(\begin{array}{c} \lambda_a \\ \mu_a
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\end{array} \right)$. The $z_a, \eta_a$ are unconstrained, and
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determine the $\lambda_a, \tilde \lambda_a$ as
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\begin{eqnarray}
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\tilde \lambda_a &=& \frac{\langle a\mi1 \, a \rangle \mu_{a\pl1} + \langle a\pl1 \, a\mi1
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\rangle \mu_a + \langle a \, a\pl1 \rangle \mu_{a\mi1}}{\langle a\mi1 \, a \rangle \langle a \, a\pl1 \rangle}, \nonumber \\
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\tilde \eta_a &=& \frac{\langle a\mi1 \, a \rangle \eta_{a\pl1} +
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\langle a\pl1 \, a\mi1 \rangle \eta_a + \langle a \, a\pl1 \rangle
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\eta_{a\mi1}}{\langle a\mi1 \, a \rangle \langle a \, a\pl1 \rangle}
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\end{eqnarray}
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where throughout
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this paper, the angle brackets $\langle \dots \rangle$ denotes
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totally antisymmetric contraction with an $\epsilon$ tensor.
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${\cal M}_{n,k}$ is cyclically invariant. It is also invariant under the little group action $(z_a, \eta_a)
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\to t_a (z_a, \eta_a)$, so $(z_a, \eta_a)$ can be taken to live in
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$\mathbb{P}^{3|4}$.
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At loop level, there is a well-defined notion of ``the integrand"
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for scattering amplitudes, which at $L$ loops is a $4L$ form. The
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loop integration variables are points in the (dual) spacetime $x^\mu_i$,
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which in turn can be associated with $L$ lines in momentum-twistor
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space that we denote as ${\cal L}_{(i)}$ for $i = 1, \cdots, L$. The
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$4L$ form is \cite{Andrewloop, LDbox, LocalIntegrand} \be {\cal
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M}(z_a, \eta_a; {\cal L}_{(i)}) \ee We can specify the line by
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giving two points ${\cal L}_{1 (i)},{\cal L}_{2 (i)}$ on it, which
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we can collect as ${\cal L}_{\gamma(i)}$ for $\gamma = 1,2$. ${\cal
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L}$ can also be thought of as a $2$ plane in 4 dimensions. In
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previous work, we have often referred to the two points on the line
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${\cal L}_1, {\cal L}_2$ as ``$AB$", and we will use this notation here
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as well.
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Dual superconformal symmetry says that ${\cal M}_{n,k}$ is invariant
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under the $SL(4|4)$ symmetry acting on $(z_a,\eta_a)$ as
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(super)linear transformations. The full symmetry of the theory is the Yangian of
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$SL(4|4)$.
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\section{Triangles $\to$ Positive Grassmannian}
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To begin with, let us start with the simplest and most familiar
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geometric object of all, a triangle in two dimensions. Thinking
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projectively, the vertices are $Z_1^I,Z_2^I,Z_3^I$ where $I=1,\dots,
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3$. The interior of the triangle is a collection of points of the
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form \be Y^I = c_1 Z_1^I + c_2 Z_2^I + c_3 Z_3^I \ee
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where we span over all $c_a$ with
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\be
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c_a > 0
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\ee
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$$
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\includegraphics[scale=.75]{pix12.pdf}
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$$
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More precisely, the interior of a triangle is associated with a
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triplet $(c_1,c_2,c_3)/GL(1)$, with all ratios $c_a/c_b > 0$, so
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that the $c_a$ are either all positive or all negative, but here and
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in the generalizations that follow, we will abbreviate this by
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calling them all positive. Including the closure of the triangle
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replaces ``positivity" with ``non-negativity", but we will continue
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to refer to this as ``positivity" for brevity.
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One obvious generalization
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of the triangle is to an $(n-1)$ dimensional simplex in a general
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projective space, a collection $(c_1, \dots, c_n)/GL(1)$, with $c_a
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> 0$. The $n$-tuple $(c_1, \dots, c_n)/GL(1)$ specifies a line in $n$
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dimensions, or a point in $\mathbb{P}^{n-1}$. We can generalize this
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to the space of $k$-planes in $n$ dimensions--the Grassmannian
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$G(k,n)$--which we can take to be a collection of $n$
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$k-$dimensional vectors modulo $GL(k)$ transformations, grouped into
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a $k \times n$ matrix \be C = \left( \begin{array}{ccc} & & \\ c_1
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& \dots & c_n \ \\ & & \end{array} \right)/GL(k) \ee
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Projective space is the special case of $G(1,n)$. The notion of
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positivity giving us the ``inside of a simplex" in projective space
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can be generalized to the Grassmannian. The only possible $GL(k)$
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invariant notion of positivity for the matrix $C$ requires us to fix
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a particular ordering of the columns, and demand that all minors in
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this ordering are positive: \be \langle c_{a_1} \dots c_{a_k}
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\rangle > 0 \, \, {\rm for} \, \, a_1<\dots<a_k \ee We can also talk
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about the very closely related space of positive matrices
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$M_+(k,n)$, which are just $k \times n$ matrices with all positive
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ordered minors. The only difference with the positive Grassmannian
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is that in $M_+(k,n)$ we are not moding out by $GL(k)$.
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Note that while both $M_+(k,n)$ and $G_+(k,n)$ are defined with a
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given ordering for the columns of the matrices, they have a natural
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cyclic structure. Indeed, if $(c_1, \dots, c_n)$ give a positive
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matrix, then positivity is preserved under the (twisted) cyclic
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action $c_1 \to c_2, \dots, c_n \to (-1)^{k-1} c_1$.
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\section{Polygons $\to$ (Tree) Amplituhedron ${\cal A}_{n,k}(Z)$}
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Another natural generalization of a triangle is to a more
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general polygon with $n$ vertices $Z^I_1, \dots, Z^I_n$. Once again
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we would like to discuss the interior of this region. However in
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general this is not canonically defined--if the points $Z_a$ are
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distributed randomly, they don't obviously enclose a region where all
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the $Z_a$ are all vertices. Only if the $Z_a$ form a {\it convex}
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polygon do we have a canonical ``interior" to talk about:
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$$
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\includegraphics[scale=.65]{pix21.pdf}
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$$
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Now, convexity for the $Z_a$ is a special case of positivity in the
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sense of the positive matrices we have just defined. The points
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$Z_a$ form a closed polygon only if the $3 \times n$ matrix with
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columns $Z_a$ has all positive (ordered) minors: \be \langle Z_{a_1}
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Z_{a_2} Z_{a_3} \rangle > 0 \quad {\rm for} \quad a_1 < a_2 < a_3
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\ee Having arranged for this, the interior of the polygon is given
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by points of the form \be Y^I = c_1 Z_1^I + c_2 Z_2^I + \dots c_n
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Z_n^I \quad {\rm with}\quad c_a > 0 \ee Note that this can be
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thought of as an interesting pairing of two different positive
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spaces. We have \be (c_1, \dots, c_n) \subset G_+(1,n),\quad
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\left(Z_1, \dots, Z_n \right) \subset M_+(3,n) \ee If we jam them
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together to produce \be Y^I = c_a Z_a^I \ee for fixed $Z_a$, ranging
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over all $c_a$ gives us all the points on the inside of the
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polygon, living in $G(1,3)$.
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This object has a natural generalization to higher projective
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spaces; we can consider $n$ points $Z_a^I$ in $G(1,1+m)$, with $I =
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1, \dots, 1+m$, which are positive \be \langle Z_{a_1} \dots Z_{a_{1
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+ m}} \rangle > 0 \ee Then, the analog of the ``inside of the
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polygon" are points of the form \be Y^I = c_a Z_a^I, \quad {\rm
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with} \quad c_a > 0 \ee This space is very closely related to the
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``cyclic polytope" \cite{cyclic}, which is the convex hull of $n$
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ordered points on the moment curve in $\mathbb{P}^{m}$, $Z_a =
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(1,t_a,t_a^2, \dots, t_a^{m})$, with $t_1 < t_2 \dots < t_n$. From
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our point of view, forcing the points to lie on the moment curve is
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overly restrictive; this is just one way of ensuring the positivity
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of the $Z_a$.
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We can further generalize this structure into the Grassmannian. We
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take positive external data as $(k + m)$ dimensional vectors
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$Z_a^I$ for $I =1, \dots, k+m$. It is natural to restrict $n \geq
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(k+m)$, so that the external $Z_a$ fill out the entire $(k+m)$
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dimensional space. Consider the space of $k$-planes in this $(k+m)$
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dimensional space, $Y \subset G(k,k+m)$, with co-ordinates \be
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Y_\alpha^I, \, \alpha = 1, \dots k, \, I = 1, \dots, k+m \ee We then
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consider a subspace of $G(k,k+m)$ determined by taking all possible
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``positive" linear combinations of the external data, \be Y = C
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\cdot Z \ee or more explicitly \be Y_\alpha^I = C_{\alpha a} Z_a^I
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\ee where \be C_{\alpha a} \subset G_+(k,n), Z_a^I \subset
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M_+(k+m,n) \ee
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It is trivial to see that this space is cyclically invariant if $m$ is even: under the twisted cyclic symmetry,
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$Z_n \to (-1)^{k+m-1} Z_1$ and $c_n \to (-1)^{k-1} c_1$, and the product is invariant for even $m$.
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|
We call this space the generalized tree amplituhedron ${\cal A}_{n,k,m}(Z)$. The polygon is the simplest case with $k=1,m=2$. Another special case is $n = (k+m)$, where we can use $GL(k+m)$
|
|
transformations to set the external data to the identity matrix
|
|
$Z_a^I = \delta_a^I$. In this case ${\cal A}_{k+m,k,m}$ is identical to the usual
|
|
positive Grassmannian $G_+(k,k+m)$.
|
|
|
|
The case of immediate relevance to physics is $m=4$, and we will refer to this as the tree amplituhedron
|
|
${\cal A}_{n,k}(Z)$. The tree amplituhedron lives in a
|
|
$4k$ dimensional space and is not trivially visualizable. For $k=1$, it
|
|
is a polytope, with inequalities determined by linear
|
|
equations, while for $k>1$, it is not a polytope and is
|
|
more ``curvy". Just to have a picture,
|
|
below we sketch a 3-dimensional face of the 4 dimensional
|
|
amplituhedron for $n=8$, which turns out to be the space $Y = c_1 Z_1 +\dots c_7
|
|
Z_7$ for $Z_a$ positive external data in $\mathbb{P}^3$:
|
|
$$
|
|
\includegraphics[scale=.6]{pix11.pdf}
|
|
$$
|
|
|
|
\section{Why Positivity?}
|
|
|
|
We have motivated the structure of the amplituhedron by mimicking
|
|
the geometric idea of the ``inside" of a convex polygon.
|
|
However there is a simpler and deeper origin of the need for
|
|
positivity. We can attempt to define $Y = C \cdot Z$ with no
|
|
positive restrictions on $C$ or $Z$. But in general, this will not
|
|
be projectively meaningful, and this expression won't allow us to
|
|
define a region in $G(k,k+m)$. The reason is that for $n > k+m$,
|
|
there is always some linear combination of the $Z_a$ which sum to
|
|
zero! We have to take care to avoid this happening, and in order to
|
|
avoid ``0" on the left hand side, we obviously need positivity
|
|
properties on both the $Z$'s and the $C$'s.
|
|
|
|
It is simple and instructive to see why positivity ensures that the
|
|
$Y = C \cdot Z$ map is projectively well-defined. We will see this
|
|
as a by-product of locating the co-dimension one boundaries of the
|
|
generalized tree amplituhedron. Let us illustrate the idea already
|
|
for the simplest case of the polygon with $k=1, m=2$, with $Y = c_1
|
|
Z_1 + \dots c_n Z_n$. In order to look at the boundaries of the
|
|
space, let us compute $\langle Y Z_i Z_j \rangle$ for some $i,j$. If
|
|
as we sweep through all the allowed $c$'s, $\langle Y Z_i Z_j
|
|
\rangle$ changes sign from being positive to negative, then
|
|
somewhere $\langle Y Z_i Z_j \rangle \to 0$ and $Y$ lies on the line
|
|
$(Z_i Z_j)$ in the interior of the space, thus $(Z_i Z_j)$ should
|
|
not be a boundary of the polygon. On the other hand, if $\langle Y
|
|
Z_i Z_j \rangle$ everywhere has a uniform sign, then $(Z_i Z_j)$ is
|
|
a boundary of the polygon:
|
|
|
|
$$
|
|
\includegraphics[scale=.6]{pix22.pdf}
|
|
$$
|
|
|
|
Of course for the polygon it is trivial to directly see that the
|
|
co-dimension one boundaries are just the lines $(Z_i Z_{i+1})$, but
|
|
we wish to see this more algebraically, in a way that will
|
|
generalize to the amplituhedron where ``seeing" is harder.
|
|
So, we compute \be \langle Y Z_i Z_j \rangle =\sum_a c_a \langle Z_a
|
|
Z_i Z_j \rangle \ee We can see why there is some hope for the
|
|
positivity of this sum, since the $c_a > 0$, and also ordered minors
|
|
of the $Z's$ are positive. It is however obvious that if $i,j$ are
|
|
not consecutive, some of the terms in this sum will be positive, but
|
|
some (where $a$ is stuck between $i,j$) will be negative. But
|
|
precisely when $i,j$ are consecutive, we get a manifestly
|
|
positive sum: \be \langle Y Z_i Z_{i+1} \rangle = \sum_a c_a \langle
|
|
Z_a Z_i Z_{i+1} \rangle > 0 \ee Since $\langle Z_a Z_i Z_{i+1}
|
|
\rangle > 0$ for $a \neq i, i+1$, this is manifestly positive. Thus
|
|
the boundaries are lines $(Z_i Z_{i+1})$ as expected.
|
|
|
|
This also tells us that the map $Y = C \cdot Z$ is projectively well-defined.
|
|
There is no way to get $Y \to 0$, since this would make the left hand side
|
|
identically zero, which is impossible without making all the $c_a$
|
|
vanish, which is not permitted as we we mod out by
|
|
$GL(1)$ on the $c_a$.
|
|
|
|
We can extend this logic to higher $k,m$. Let's look at the case
|
|
$m=4$ already for $k=1$.
|
|
We can investigate whether the plane $(Z_i Z_j Z_k Z_l)$ is a boundary by computing
|
|
\be \langle Y Z_i Z_j Z_k Z_l \rangle = \sum_a c_a \langle Z_a Z_i
|
|
Z_j Z_k Z_l \rangle \ee Again, this is not in general positive. Only for
|
|
$(i,j,k,l)$ of the form $(i,i+1,j, j+1)$, we have \be \langle Y Z_i
|
|
Z_{i+1} Z_j Z_{j+1} \rangle = \sum_a c_a \langle Z_a Z_i Z_{ i+1}
|
|
Z_j Z_{j+1} \rangle > 0 \ee For general even the $m$, the
|
|
boundaries are when $Y$ is on the plane\\ $(Z_{i} Z_{i+1}
|
|
\dots Z_{i_{m/2 - 1}} Z_{i_{m/2}})$. This again shows that the $Y =
|
|
C \cdot Z$ is projectively well-defined. The result extends
|
|
trivially to general $k$, provided the positivity of $C$ is
|
|
respected. For $m = 4$ the boundaries are again when the $k$-plane $(Y_1 \cdots Y_k)$ is on $(Z_i Z_{i+1} Z_j
|
|
Z_{j+1})$, as follows from \be \langle Y_1 \dots Y_k Z_i Z_{i+1} Z_j
|
|
Z_{j+1} \rangle = \sum_{a_1< \dots < a_k} \langle c_{a_1} \dots
|
|
c_{a_k} \rangle \langle Z_{a_1} \dots Z_{a_k} Z_i Z_{i+1} Z_j
|
|
Z_{j+1} \rangle > 0 \ee which also shows that $Y$ is always a full rank
|
|
$k$-plane in $k+4$ dimensions.
|
|
|
|
The emergence of boundaries on the plane $(Z_{i} Z_{i+1} Z_j
|
|
Z_{j+1})$ is a simple and striking consequence of positivity. We will
|
|
shortly understand that the location of these boundaries are the
|
|
``positive origin" of locality from the geometry of the
|
|
amplituhedron.
|
|
|
|
\section{Cell Decomposition}
|
|
|
|
The tree amplituhedron can be thought of as the image of the
|
|
top-cell of the the positive Grassmannian $G_+(k,n)$ under the map
|
|
$Y = C \cdot Z$. Since ${\rm dim}\, G(k,k+m) = m k \leq\,{\rm
|
|
dim}\,G(k,n)= k (n -k)$ for $n \geq k+m$, this is in general a
|
|
highly redundant map. We can already see this in the simplest case
|
|
of the polygon, which lives in 2 dimensions, while the $c_a$ span an
|
|
$(n-1)$ dimensional space. The non-redundant maps into $G(k,k+m)$
|
|
can only come from the $m \times k$ dimensional ``cells" of
|
|
$G_+(k,n)$. For the polygon, these are the cells we can label as
|
|
$(i,j,k)$, where all but $(c_i,c_j,c_k)$ are non-vanishing. The
|
|
image of these cells in the $Y$-space are of course just the
|
|
triangles with vertices at $Z_i,Z_j,Z_k$, which lie inside the
|
|
polygon.
|
|
|
|
The union of all these triangles covers the inside of the polygon.
|
|
However, we can also cover the inside of the polyon more nicely with
|
|
non-overlapping triangles, giving a triangulation. Said in a
|
|
heavy-handed way, we find a collection of 2 dimensional cells of
|
|
$G_+(1,n)$, so that their images in $Y$ space are non-overlapping
|
|
except on boundaries, and collectively cover the entire polygon. Of
|
|
course these collections of cells are not unique--there are many
|
|
different triangulations of the polygon. A particularly simple one
|
|
is
|
|
$$
|
|
\includegraphics[scale=.6]{pix23.pdf}
|
|
$$
|
|
which we can write as \be \sum_i (1\,i\,i\pl1) \ee Sticking with
|
|
$k=1$ but moving to $m=4$, the four-dimensional cells of $G_+(1,n)$
|
|
are labeled by five non-vanishing $c$'s $(c_i,c_j,c_k,c_l,c_m)$.
|
|
While it is harder to visualize, one can easily show algebraically
|
|
that the above simple triangulation of the polygon generalizes to
|
|
\be \sum_{i<j} (1\,i\,i\pl1\,j\,j\pl1) \ee
|
|
|
|
This expression is immediately recognizable to physicists familiar
|
|
with scattering amplitudes in ${\cal N} = 4$ SYM. If the
|
|
$(i,j,k,l,m)$ are interpreted as ``R-invariants", this expression is
|
|
one of the canonical BCFW representations of the $k=1$ ``NMHV" tree
|
|
amplitudes. In the positive Grassmannian representation for
|
|
amplitudes \cite{N4,N6}, R-invariants are precisely associated with
|
|
the corresponding four-dimensional cells of $G(1,n)$.
|
|
|
|
For general $k$, $m$ any $m \times k$ dimensional cell of $G_+(k,n)$
|
|
maps under $Y = C \cdot Z$ into some region or cell in $G(k,k+m)$.
|
|
Said more explicitly, consider an $m \times k$ dimensional cell
|
|
$\Gamma$ of the $G_+(k,n)$, with ``positive co-ordinates"
|
|
$C^\Gamma(\alpha^\Gamma_1, \dots, \alpha^\Gamma_{m \times k})$ \cite{N6}.
|
|
Putting $Y = C(\alpha) \cdot Z$ and scanning over all positive
|
|
$\alpha$'s, this carves out a region in $G(k,k+m)$ which is a
|
|
corresponding cell $\Gamma$ of the tree amplituhedron. A cell
|
|
decomposition is a collection $T$ of non-overlapping cells $\Gamma$
|
|
which cover the entire amplituhedron.
|
|
|
|
The case of immediate relevance for physics is $m=4$. For any $k$,
|
|
the BCFW decomposition of tree amplitudes gives us a collection of
|
|
$4 \times k$ dimensional cells of the positive Grassmannian. We have
|
|
performed extensive checks for high $k$ and $n$, that for positive
|
|
external $Z$, under $Y = C \cdot Z$ these cells are beautifully
|
|
mapped into non-overlapping regions of $G(k,k+4)$ that together
|
|
cover the entire tree amplituhedron. As we have stressed, other than
|
|
the desire to make the final result local and unitary, we did not
|
|
previously have a rational for thinking about this particular
|
|
collection of cells of $G_+(k,n)$. Now we know what natural question
|
|
this collection of cells are answering: taken together they
|
|
``cellulate" the tree amplituhedron. We will shortly see how to
|
|
directly associate the amplitude itself directly with the geometry
|
|
of the amplituhedron.
|
|
|
|
|
|
\section{``Volume" as Canonical Form}
|
|
|
|
Before discussing how to determine the (super)amplitude from the
|
|
geometry, let us define the notion of a ``volume" associated with
|
|
the amplituhedron. As should by now be expected, we will merely generalize a simple
|
|
existing idea from the world of triangles and polygons.
|
|
|
|
The usual notion of ``area" has units and is obviously not projectively
|
|
meaningful. However there is a closely related idea that is. For the
|
|
triangle, we can consider a rational 2-form in $Y$-space, which has
|
|
logarithmic singularities on the boundaries of the triangle. This is
|
|
naturally associated with positive co-ordinates for the triangle, if
|
|
we expand $Y = Z_3 + \alpha_1 Z_1 + \alpha_2 Z_2$, then the form is \be
|
|
\Omega_{123} = \frac{d\alpha_1}{\alpha_1} \frac{d\alpha_2}{\alpha_2} \ee which can be
|
|
re-written more invariantly as \be \Omega_{123} = \frac{\langle Y d
|
|
Y d Y \rangle \langle 1 2 3 \rangle^2}{\langle Y 12 \rangle \langle
|
|
Y 2 3 \rangle \langle Y 3 1 \rangle} \ee We can extend this to a
|
|
form $\Omega_P$ for the convex polygon $P$. The defining property of
|
|
$\Omega_P$ is that
|
|
|
|
\begin{center}
|
|
$\Omega_P$ has logarithmic singularities on all the boundaries of
|
|
$P$.
|
|
\end{center}
|
|
$\Omega_P$ can be obtained by first triangulating the polygon in
|
|
some way, then summing the elementary two-form for each triangle,
|
|
for instance as \be \Omega_P = \sum_i \Omega_{1\,i\,i\pl1}. \ee Each
|
|
term in this sum has singularities corresponding to $Y$ hitting the
|
|
boundaries of the corresponding triangle. Most of these singularities, associated
|
|
with the internal edges of the triangulation, are spurious, and
|
|
cancel in the sum. Of course the full form $\Omega_P$ is independent
|
|
of the particular triangulation.
|
|
|
|
This form is closely related to an area, not directly of the polygon
|
|
$P$, but its dual $\tilde{P}$, which is also a convex polygon
|
|
\cite{N7}. If we dualize so that points are mapped to lines and
|
|
lines to points, then a point $Y$ {\it inside} $P$ is mapped to a
|
|
line $Y$ {\it outside} $\tilde{P}$. If we write $\Omega_P = \langle
|
|
Y d^2 Y \rangle V(Y)$, then $V(Y)$ is the area of $\tilde{P}$ living
|
|
in the euclidean space defined by $Y$ as the line at infinity.
|
|
|
|
This form can be generalized to the tree amplituhedron in an obvious
|
|
way. We define a rational form $\Omega_{n,k}(Y;Z)$ with the property
|
|
that
|
|
|
|
\begin{center}
|
|
$\Omega_{n,k}(Y;Z)$ has logarithmic singularities on all the
|
|
boundaries of ${\cal A}_{n,k}(Z)$.
|
|
\end{center}
|
|
|
|
Just as for the polygon, one concrete way of computing $\Omega$ is
|
|
to give a cell decomposition of the amplituhedron. For any cell
|
|
$\Gamma$ associated with positive co-ordinate $(\alpha^\Gamma_1,
|
|
\dots, \alpha^\Gamma_{4k})$, there is an associated form with
|
|
logarithmic singularities on the boundaries of the cell \be
|
|
\Omega_{n,k}^\Gamma(Y; Z) = \prod_{i = 1}^{4k} \frac{d
|
|
\alpha_i^\Gamma}{\alpha_i^\Gamma} \ee For instance, consider 4
|
|
dimensional cells for $k=1$, associated with cells in $G_+(1,n)$
|
|
which are vanishing for all but columns $a_1, \dots, a_5$, with
|
|
positive co-ordinates\\ $(\alpha_{a_1}, \dots, \alpha_{a4},
|
|
\alpha_{a_5} =1)$. Its image in $Y$ space is simply \be Y =
|
|
\alpha_{a1} Z_{a_1} + \dots \alpha_{a_4} Z_{a_4} + Z_{a_5} \ee and
|
|
the form is \be \frac{d \alpha_{a1}}{\alpha_{a1}} \dots \frac{d
|
|
\alpha_{a4}}{\alpha_{a4}} = \frac{\langle Y d^4 Y \rangle \langle
|
|
Z_{a_1} Z_{a_2} Z_{a_3} Z_{a_4} Z_{a_5} \rangle^4}{\langle Y Z_{a_1}
|
|
Z_{a_2} Z_{a_3} Z_{a_4} \rangle \dots \langle Y Z_{a_5} Z_{a_1}
|
|
Z_{a_2} Z_{a_3} \rangle} \ee Now, given a collection of cells $T$
|
|
that cover the full amplituhedron, $\Omega_{n,k}(Y;Z)$ is given by
|
|
\be \Omega_{n,k}(Y;Z) = \sum_{\Gamma \subset T}
|
|
\Omega_{n,k}^\Gamma(Y;Z) \ee As with the polygon, the form is
|
|
independent of the particular cell decomposition.
|
|
|
|
Note that the definition of the amplituhedron itself crucially
|
|
depends on the positivity of the external data $Z$, and this
|
|
geometry in turn determines the form $\Omega$. However, once this
|
|
form is in hand, it can be analytically continued to any
|
|
(complex!) $Y$ and $Z$.
|
|
|
|
\section{The Superamplitude}
|
|
|
|
We have already defined central objects in our story: the
|
|
tree amplituhedron, together with the associated form $\Omega$
|
|
that is loosely speaking its ``volume". The
|
|
tree super-amplitude ${\cal M}_{n,k}$ can be
|
|
directly extracted from $\Omega_{n,k}(Z)$. To see how, note that
|
|
we we can always use a $GL(4 +k)$ transformation to send $Y \to Y_0$
|
|
as \be Y_0 = \left(\begin{array}{ccc} & 0_{4 \times k} & \\
|
|
\hdashline & 1_{k \times k} & \end{array} \right) \ee We can think
|
|
of the 4 dimensional space complementary to $Y_0$, acted on by an
|
|
unbroken $GL(4)$ symmetry, as the usual $\mathbb{P}^3$ of
|
|
momentum-twistor space. Accordingly, we identify the top four
|
|
components of the $Z_a$ with the usual bosonic momentum-twistor
|
|
variables $z_a$: \be Z_a = \left(\begin{array}{c} z_a \\ *_1 \\
|
|
\vdots \\ *_k
|
|
\end{array}\right) \ee We still have to decide how to interpret the
|
|
remaining $k$ entries of $Z_a$. Clearly, if they are normal bosonic
|
|
variables, we have an infinite number of extra degrees of freedom.
|
|
It is therefore natural to try and make the remaining components
|
|
infinitesimal, by saying that some ${\cal N} + 1$'st power of them
|
|
vanishes. This is equivalent to saying that each entry can be
|
|
written as \be Z_a = \left(\begin{array}{c} z_a \\ \phi^A_1 \cdot
|
|
\eta_{1 A} \\ \vdots \\ \phi^A_k \cdot \eta_{A k} \end{array}\right)
|
|
\ee where $\phi_{1,\dots,k}$ and $\eta_a$ are Grassmann parameters,
|
|
and $A = 1, \dots, {\cal N}$.
|
|
|
|
Now there is a unique way to extract the amplitude. We simply
|
|
localize the form $\Omega_{n,k}(Y;Z)$ to $Y_0$, and integrate over the
|
|
$\phi$'s: \be {\cal M}_{n,k}(z_a,\eta_a) = \int d^{\cal N} \phi_1 \dots
|
|
d^{\cal N} \phi_k \int \Omega_{n,k}(Y;Z) \delta^{4k}(Y;Y_0)
|
|
\label{super}\ee Here $\delta^{4k}(Y;Y_0)$ is a projective $\delta$
|
|
function \be \delta^{4k}(Y;Y_0) = \int d^{k \times k}
|
|
\rho_\alpha^\beta \, {\rm det}(\rho)^4 \, \delta^{k \times
|
|
(k+4)}(Y_\alpha^I - \rho_{\alpha}^\beta Y_{0 \beta}^I) \ee Note that
|
|
there is really no integral to perform in the second step; the delta
|
|
functions fully fix $Y$. Any form on $G(k, k+4)$ is of the form
|
|
\begin{equation}
|
|
\Omega = \langle Y_1 \dots Y_k d^4 Y_1 \rangle \dots \langle Y_1
|
|
\dots Y_k d^4 Y_k \rangle \times \omega_{n,k}(Y;Z)
|
|
\end{equation}
|
|
and our expression just says that \be {\cal M}_{n,k}(z_a,\eta_a) = \int
|
|
d^{\cal N} \phi_1 \dots d^{\cal N} \phi_k \omega_{n,k}(Y_0;Z_a) \ee
|
|
Note that we can define this operation for any ${\cal N}$, however,
|
|
only for ${\cal N} = 4$ does ${\cal M}_{n,k}$ further have weight zero under the
|
|
rescaling $(z_a,\eta_a)$.
|
|
|
|
This connection between the form and the super-amplitude also allows
|
|
us to directly exhibit the relation between our super-amplitude
|
|
expressions and the Grassmannian formulae of \cite{N4,N6}. Consider
|
|
the form in $Y$-space associated with a given $4k$ dimensional cell
|
|
$\Gamma$ of $G_+(k,n)$. Then, if $C^\Gamma_{\alpha
|
|
a}(\alpha_1,\dots, \alpha_{4k})$ are positive co-ordinates for the
|
|
cell, and $\Omega^\Gamma = \frac{d\alpha^\Gamma_1}{\alpha^\Gamma_1}
|
|
\dots \frac{d\alpha^\Gamma_{4k}}{\alpha^\Gamma_{4k}}$ is the
|
|
associated form in $Y$ space, then it is easy to show that \be \int
|
|
d^4 \phi_1 \dots d^4 \phi_k \int \Omega^\Gamma \delta^{4k}(Y;Y_0) =
|
|
\int \frac{d\alpha^\Gamma_1}{\alpha^\Gamma_1} \dots
|
|
\frac{d\alpha^\Gamma_{4k}}{\alpha^\Gamma_{4k}}
|
|
\delta^{4k|4k}(C_{\alpha a}(z) {\cal Z}_a) \ee where ${\cal Z}_a =
|
|
(z_a|\eta_a)$ are the super momentum-twistor variabes. This is
|
|
precisely the formula for computing on-shell diagrams (in
|
|
momentum-twistor space) as described in \cite{N4,LD1,N6}. Thus,
|
|
while the amplituhedron geometry and the associated form $\Omega$
|
|
are purely bosonic, we have extracted from them super-amplitudes
|
|
which are manifestly supersymmetric. Indeed, the connection to the
|
|
Grassmannian shows much more--the superamplitude obtained for each
|
|
cell is manifestly Yangian invariant \cite{N6}.
|
|
|
|
\section{Hiding Particles $\to$ Loop Positivity in $G_+(k,n;L)$}
|
|
The direct generalization of ``convex polygons" into the
|
|
Grassmannian $G(k,k+4)$ has given us the tree amplituhedron. We will
|
|
now ask a simple question: can we ``hide particles" in a natural
|
|
way? This will lead to an extended notion of positivity giving us
|
|
loop amplitudes.
|
|
|
|
It is trivial to imagine what we might mean by hiding a single
|
|
particle, but as we will see momentarily, the idea of hiding
|
|
particles is only natural if we hide {\it pairs} of {\it adjacent}
|
|
particles. To pick a concrete example, suppose we have some positive
|
|
matrix $C$ with columns we'll label $(A_1,B_1, 1, 2, \dots, m, A_2, B_2,
|
|
m+1, \dots n)$. We can always gauge-fix the $A_1,B_1$ and $A_2,B_2$ columns
|
|
so that the matrix takes the form
|
|
|
|
$$\bordermatrix{\text{}&A_1&B_1&1&2&\ldots&m&A_2&B_2&m+1&\ldots&n\cr
|
|
&1&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
|
|
&0&1&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
|
|
&0&0&\ast&\ast&\ldots&\ast&1&0&\ast&\ldots & \ast\cr
|
|
&0&0&\ast&\ast&\ldots&\ast&0&1&\ast&\ldots & \ast\cr
|
|
&0&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr
|
|
&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr
|
|
&0&0&\ast&\ast&\ldots&\ast&0&0&\ast&\ldots & \ast\cr}$$
|
|
We would now like to ``hide" the particles $A_1,B_1,A_2,B_2$. We do
|
|
this simply by chopping out the corresponding columns. The remaining
|
|
matrix can be grouped into the form \be \left(\begin{array}{ccc}&
|
|
D_{(1)} & \\ \hdashline & D_{(2)} & \\ \hdashline & C & \end{array}
|
|
\right) \ee But the ``hidden" particles leave an echo in the
|
|
positivity properties of this matrix. The positivity of the minors
|
|
involving all of $(A_1,B_1,A_2,B_2)$, $(A_2,B_2)$ and $(A_1,B_1)$
|
|
individually, as well those not involving $A_1,B_1,A_2,B_2$ at all
|
|
enforce that the ordered maximal minors of the following matrices
|
|
\be \left(\begin{array}{ccc} & C & \end{array} \right), \left(
|
|
\begin{array}{ccc}& D_{(1)}& \\ \hdashline & C & \end{array} \right),
|
|
\left( \begin{array}{ccc} &D_{(2)}& \\ \hdashline & C & \end{array}
|
|
\right), \left(\begin{array}{ccc} & D_{(1)}& \\ \hdashline &
|
|
D_{(2)}&
|
|
\\ \hdashline &C& \end{array} \right) \ee are all positive.
|
|
|
|
We can now see why particles are most naturally hidden in pairs. If
|
|
we had instead hidden single particles as $A_1, A_2, A_3, \dots$ in
|
|
separate columns, the remaining minors would be positive or negative
|
|
depending on the orderings of $A_1,A_2,A_3, \dots$, which is
|
|
additional structure over and above the cyclic ordering of the
|
|
external data. In order to avoid this arbitrariness, we should hide
|
|
particles in even numbers, with pairs the minimal case. In order to
|
|
ensure that only minors involving the pairs $(A_i B_i)$ are taken into
|
|
account, we mod out by the $GL(2)$ action rotating the $(A_i,B_i)$ columns
|
|
into each other.
|
|
|
|
This ``hidden particle" picture has thus motivated an extended
|
|
notion of positivity associated with the Grassmannian. We are used
|
|
to considering a $k$-plane in $n$ dimensions $C$, with all ordered
|
|
minors positive. But we can also imagine a collection of $L$
|
|
2-planes $D_{(i)}$ in the $(n-k)$ dimensional complement of $C$,
|
|
schematically
|
|
$$
|
|
\includegraphics[scale=.65]{pix3.pdf}
|
|
$$
|
|
We call this space $G(k,n;L)$, and we will denote the collection of
|
|
$(D_{(i)}, C)$ as ${\cal C}$. We can extend the notion of
|
|
positivity to $G(k,n;L)$ by demanding that not only the ordered
|
|
minors of $C$, but also of $C$ with any collection of the $D_{(i)}$,
|
|
are positive. (All minors must include the matrix $C$, since the
|
|
$D_{(i)}$ are defined to live in the complement of $C$). Note that
|
|
this notion is completely permutation invariant in the $D_{(i)}$.
|
|
|
|
Very interestingly, it turns out that while we motivated this notion
|
|
of positivity by hiding particles from an underlying positive
|
|
matrix, there are positive configurations of ${\cal C}$ that can not
|
|
be obtained by hiding particles from a positive matrix in this way.
|
|
|
|
Extending the map $Y = C . Z$ in the obvious way to include the
|
|
$D$'s leads us to define the full amplituhedron.
|
|
|
|
\section{The Amplituhedron ${\cal A}_{n,k,L}(Z)$}
|
|
|
|
We can now give the full definition of the amplituhedron ${\cal
|
|
A}_{n,k,L}(Z)$. First, the external data for $n \geq k+4$
|
|
particles is given by the vectors $Z_a^I$ living in a $(4 + k)$
|
|
dimensional space; where $a= 1, \dots, n$ and $I = 1, \dots, 4 + k$.
|
|
The data is positive \be \langle Z_{a_1} \dots Z_{a_{4 + k}} \rangle
|
|
> 0 \quad {\rm for} \quad a_1 < \dots < a_{4 + k} \ee The amplituhedron
|
|
lives in $G(k,k+4;L)$: the space of $k$ planes $Y$ in $(k+4)$
|
|
dimensions, together with $L$ 2-planes ${\cal L}_{(i)}$ in the 4
|
|
dimensional complement of $Y$, schematically
|
|
$$
|
|
\includegraphics[scale=.65]{pix28.pdf}
|
|
$$
|
|
We will denote the collection of $({\cal L}_{(i)}, Y)$ as ${\cal
|
|
Y}$.
|
|
|
|
The amplituhedron ${\cal A}_{n,k,L}(Z)$ is the subspace of
|
|
$G(k,k+4;L)$ consisting of all ${\cal Y}$'s which are a positive
|
|
linear combination of the external data, \be {\cal Y} = {\cal C}
|
|
\cdot Z \ee More explicitly in components, the $k$-plane is
|
|
$Y_{\alpha}^I$, and the 2-planes are ${\cal L}_{\gamma (i)}^I$,
|
|
where $\gamma = 1,2$ and $i = 1, \dots, L$ . The amplituhedron is
|
|
the space of all $Y, {\cal L}_{(i)}$ of the form \be Y_{\alpha}^I =
|
|
C_{\alpha a} Z_a^I, \, \, {\cal L}_{\gamma (i)}^{I} = D_{\gamma a
|
|
(i)} Z_a^I \ee where as in the previous section the $C_{\alpha a}$
|
|
specifies a $k$-plane in $n$-dimensions, and the $D_{\gamma a (i)}$
|
|
are $L$ 2-planes living in the $(n-k)$ dimensional complement of
|
|
$C$, with the positivity property that for any $0 \leq l \leq L$,
|
|
all the ordered $(k + 2l) \times (k+2l)$ minors of the $(k + 2
|
|
l) \times n$ matrix \be \left(\begin{array}{ccc} & D_{(i_1)} & \\
|
|
\hdashline & \vdots & \\ \hdashline & D_{(i_l)} & \\ \hdashline & C
|
|
\end{array} \right) \ee are positive.
|
|
|
|
The notion of cells, cell decomposition and canonical form
|
|
can be extended to the full amplituhedron. A cell $\Gamma$ is
|
|
associated with a set of positive co-ordinates $\alpha^\Gamma =
|
|
(\alpha^\Gamma_1, \dots, \alpha^\Gamma_{4(k + L)})$, rational in the ${\cal C}$, such that for $\alpha$'s positive,
|
|
${\cal C}(\alpha) = (D_{(i)}(\alpha), C(\alpha))$ is in $G_+(k,n;L)$. A
|
|
cell decomposition is a collection $T$ of non-intersecting cells
|
|
$\Gamma$ whose images under ${\cal Y} = {\cal C} \cdot Z$ cover the
|
|
entire amplituhedron. The rational form $\Omega_{n,k,L}({\cal Y}; Z)$ is
|
|
defined by having the property that
|
|
|
|
\begin{center}
|
|
$\Omega_{n,k,L}(Y;Z)$ has logarithmic singularities on all the
|
|
boundaries of ${\cal A}_{n,k,L}(Z)$
|
|
\end{center}
|
|
|
|
A concrete formula follows from a cell decomposition as \be
|
|
\Omega_{n,k,L}({\cal Y}; Z) = \sum_{\Gamma \subset T} \prod_{i =
|
|
1}^{4(k + L)} \frac{d \alpha_i^\Gamma}{\alpha_i^\Gamma} \ee Of course any cell
|
|
decomposition gives the same form $\Omega_{n,k,L}$.
|
|
|
|
|
|
\section{The Loop Amplitude Form}
|
|
|
|
We can extract the $4L$-form for the loop integrand from
|
|
$\Omega_{n,k,L}$ in the obvious way. The 2-planes ${\cal L}_{(i)}$,
|
|
being in the complement of $Y_0$, can be taken to be non-vanishing
|
|
in the first 4 entries ${\cal L}^I_{ (i)} = ({\cal L}_{(i) 2 \times
|
|
4} | 0_{2 \times k})$. Each ${\cal L}_{\gamma (i)}$ gives us a line
|
|
$({\cal L}_{\gamma =1} {\cal L}_{\gamma = 2})_{(i)}$ (which we have
|
|
also been calling $(AB)_{(i)}$) in $\mathbb{P}^3$. These are the
|
|
momentum-twistor representation of the loop integration variables.
|
|
The analog of equation (\ref{super}) for the loop integrand form is
|
|
\be {\cal M}_{n,k}(z_a,\eta_a; {\cal L}_{(\gamma (i)}) = \int d^{4}
|
|
\phi_1 \dots d^4 \phi_k \int \Omega_{n,k,L}(Y,{\cal L}_{\gamma
|
|
(i)};Z) \delta^{4k}(Y;Y_0) \ee Any form on $G(k,k+4k;L)$ can be
|
|
written as
|
|
\begin{equation}
|
|
\Omega = \langle Yd^4 Y_1 \rangle \dots \langle Yd^4 Y_k \rangle
|
|
\prod_{i=1}^L \langle Y {\cal L}_{1(i)} {\cal L}_{2(i)} d^2 {\cal
|
|
L}_{1(i)} \rangle \langle Y {\cal L}_{1(i)} {\cal L}_{2(i)} d^2
|
|
{\cal L}_{2(i)} \rangle \times \omega_{n,k,L}(Y,{\cal L}_{(i)})(Z)
|
|
\end{equation}
|
|
where we denoted $Y=Y_1\dots Y_k$. So we have for the integrand of
|
|
the all-loop amplitude
|
|
\begin{align}
|
|
{\cal M}_{n,k}(z_a,\eta_a,{\cal L}_{\gamma(i)}) &= \int d^4 \phi_1
|
|
\dots d^4 \phi_k \prod_{i=1}^L \langle {\cal L}_{1(i)} {\cal
|
|
L}_{2(i)} d^2 {\cal L}_{1(i)} \rangle \langle {\cal L}_{1(i)} {\cal
|
|
L}_{2(i)} d^2 {\cal L}_{2(i)} \rangle \omega_{n,k}(Y_0,{\cal
|
|
L}_{\gamma(i)};Z_a)
|
|
\end{align}
|
|
Already the simplest case $k=0$ of the amplituhedron is interesting
|
|
at loop level. At 1-loop, we have a 2-plane in 4 dimensions $AB$,
|
|
and the $D$ matrix is just restricted to be in $G_+(2,n)$. It is
|
|
easy to see that the 4 dimensional cells of $G_+(2,n)$ are labeled
|
|
by a pair of triples $[a,b,c;x,y,z]$, where the top row of the
|
|
matrix is non-zero in the columns $(a,b,c)$ and the bottom in
|
|
columns $(x,y,z)$. A simple collection of these \be \sum _{i<j}
|
|
[1\,i\,i\pl1;\,1\,j\,j\pl1] \ee beautifully covers the amplituhedron
|
|
in this case. The map into $G(2,4)$ for each cell is \be A = Z_1 +
|
|
\alpha_i Z_i + \alpha_{i+1} Z_{i+1}, \, B = -Z_1 + \alpha_j Z_j +
|
|
\alpha_{j+1} Z_{j+1} \ee and so the form associated with the cell is
|
|
\begin{align} &\frac{d
|
|
\alpha_i}{\alpha_i} \frac{d\alpha_{i+1}}{\alpha_{i+1}}
|
|
\frac{\alpha_j}{\alpha_j} \frac{d \alpha_{j+1}}{\alpha_{j+1}} =
|
|
\frac{\langle AB d^2 A \rangle \langle AB d^2 B \rangle \langle AB
|
|
(1\,i\,i\pl1) \cap (1\,j\,j\pl1) \rangle^2}{\langle AB\,1\,i \rangle
|
|
\langle AB\,1\,i\pl1 \rangle \langle AB\,i\,i\pl1 \rangle \langle
|
|
AB\,1\,j\rangle \langle AB\,1\,j\pl1 \rangle \langle AB\,j\,j\pl1
|
|
\rangle}
|
|
\end{align}
|
|
The form $\Omega$ gives exactly the ``Kermit" expansion for the MHV
|
|
integrand given in \cite{N5}, now obtained without any reference to tree
|
|
amplitudes, forward limits or recursion relations.
|
|
|
|
In this simple case, direct triangulation of the space is straightforward. But we could also have worked backwards, starting with the BCFW formula, and recognizing how each term in the ``Kermit" expansion is associated with positive co-ordinates for some cell of the amplituhedron. We could then observe that, remarkably, these cells are non-overlapping, and together cover the full amplituhedron.
|
|
|
|
In order to illustrate more of the structure of the loop
|
|
amplituhedron, including the interplay between the $``C"$ and $``D"$
|
|
matrices, let us consider the 1-loop $k=1$ amplitude for $n=6$.
|
|
There are 16 terms in the BCFW recursion, which can all be mapped
|
|
back to their $Y, AB$ space form, and in turn associated with
|
|
positive co-ordinates in the amplituhedron. For instance, one of
|
|
BCFW terms is
|
|
|
|
$$
|
|
\frac{\la YAB13\ra\la YAB(561)\cap(2345)\ra^4\la
|
|
YAB(123)\cap(Y456)\ra^2}{\begin{array}{c} \la Y2345\ra\la
|
|
YAB(561)\cap(Y345)\ra\la YAB(561)\cap(Y234)\ra\la
|
|
YAB(561)\cap(Y235)\ra\la YAB56\ra \\ \la
|
|
YAB(561)\cap(Y45(23)\cap(YAB1))\ra\la YAB12\ra\la YAB23\ra\la
|
|
YAB13\ra\la YAB15\ra\la YAB16\ra\end{array}}
|
|
$$
|
|
While it may not be immediately apparently, this is nothing but the
|
|
``dlog" canonical form associated with the following positive
|
|
co-ordinates for the $(D,C)$ matrix
|
|
$$
|
|
\left(\begin{array}{ccc} & D & \\ \hdashline & C & \end{array}
|
|
\right) = \left(
|
|
\begin{array}{cccccc}
|
|
1 & x & y & 0 & 0 & 0 \\
|
|
-w & 0 & 0 & 0 & -1 & -z \\ \hdashline
|
|
w & xt_1 & t_2+t_1y & t_3 & 1+t_4 & z \\
|
|
\end{array}
|
|
\right)
|
|
$$
|
|
This exercise can be repeated with all $16$ BCFW terms. The
|
|
corresponding $(D,C)$ matrices are
|
|
$$
|
|
\hspace{-0.5cm}\includegraphics[scale=1]{six_nmhv.pdf}
|
|
$$
|
|
One can easily check that for all variables positive, the bottom row
|
|
of these matrices is positive, and all the ordered $3 \times 3$
|
|
minors are also positive. For any cell, we can range over all the
|
|
positive variables, which under the ${\cal Y} = {\cal C} \cdot Z$
|
|
gives an image of the cell in $(Y,AB)$ space. Remarkably, we find
|
|
that these cells are non-overlapping, and cover the entire space.
|
|
This can be checked directly in a simple way. We begin by fixing
|
|
positive external data $(Z_1, \cdots, Z_6)$. We then choose any
|
|
positive matrix ${\cal C}$ at random, which gives an associated
|
|
point ${\cal Y}$ inside the amplituhedron. We can ask whether or not
|
|
this point is contained in one of the cells, by seeing whether
|
|
${\cal Y}$ can be reproduced with positive values for all eight
|
|
variables of the cell. Doing this we find that every point in the
|
|
amplituhedron is contained in just one of these cells (except of
|
|
course for points on the common boundaries of different cells). The
|
|
cells taken together therefore give a cellulation of the
|
|
amplituhedron.
|
|
|
|
Note that the form shown above, associated with a BCFW term, has some
|
|
physical poles (like $\langle Y AB 12 \rangle$), but also many
|
|
unphysical poles. The unphysical poles are associated with
|
|
boundaries of the cell that are ``inside" the amplituhedron, and not
|
|
boundaries of the amplituhedron themselves. These boundaries are
|
|
spurious, and so are the corresponding poles, which cancel in the
|
|
sum over all BCFW terms.
|
|
|
|
We have checked in many other examples, for higher $k$ and also at
|
|
higher loops, that $(a)$ BCFW terms can be expressed as canonical
|
|
forms associated with cells of the amplituhedron and $(b)$ these
|
|
collection of cells do cover the amplituhedron.
|
|
|
|
It is satisfying to have a definition of the loop amplituhedron that
|
|
lives directly in the space relevant for loop amplitudes. This is in
|
|
contrast with the approach to computing the loop integrand using
|
|
recursion relations, which ultimately traces back to higher $k$ and
|
|
$n$ tree amplitudes. Consider the simple case of the 2-loop
|
|
4-particle amplitude. We are after a form in the space of two
|
|
2-planes $(AB)_1, (AB)_2$ in four dimensions. The BCFW approach begins with
|
|
the $k=2, n=8$ tree amplitudes, and arrives at the form we are
|
|
interested in after taking two ``forward limits". But the
|
|
amplituhedron lives directly in the $(AB)_1, (AB)_2$ space, and we can find
|
|
a cell decomposition for it directly, yielding the form without
|
|
having to refer to any tree amplitudes.
|
|
|
|
We have understood how to directly ``cellulate" the amplituhedron in
|
|
a number of other examples, and strongly suspect that there will be
|
|
a general understanding for how to do this. The BCFW
|
|
decomposition of tree amplitudes seems to be associated with
|
|
particularly nice, canonical cellulations of the tree
|
|
amplituhedron. Loop level BCFW also gives a cell decomposition. The ``direct" cellulations we have
|
|
found in many cases are however simpler, without an obvious connection to the BCFW expansion.
|
|
|
|
|
|
\section{Locality and Unitarity from Positivity}
|
|
|
|
Locality and unitarity are encoded in the positive geometry of the
|
|
amplituhedron in a beautiful way. As is well-known, locality and
|
|
unitarity are directly reflected in the singularity structure of the
|
|
integrand for scattering amplitudes. In momentum-twistor language,
|
|
the only allowed singularities at tree-level should occur when
|
|
$\langle Z_i Z_{i+1} Z_j Z_{j+1} \rangle \to 0$; in the loop-level
|
|
integrand, we can also have poles of the form $\langle AB\,i\,i\pl1
|
|
\rangle \to 0$, and $\langle AB_{(i)} AB_{(j)} \rangle \to 0$.
|
|
Unitarity is reflected in what happens as poles are approached,
|
|
schematically we have \cite{N6}
|
|
$$
|
|
\includegraphics[scale=.8]{pix30.pdf}
|
|
$$
|
|
|
|
Given the connection between the form $\Omega_{n,k,L}$ and the
|
|
amplitude, it is obvious that the first (co-dimension one) poles of
|
|
the amplitude are associated with the co-dimension one ``faces" of
|
|
the amplituhedron. For trees, we have already seen that, remarkably,
|
|
positivity forces these faces to be precisely where $\langle Y_1
|
|
\dots Y_k Z_i Z_{i+1} Z_j Z_{j+1} \rangle \to 0$, exactly as needed
|
|
for locality. The analog statement for the full loop amplituhedron
|
|
also obviously includes $\langle Y_1 \cdots Y_k AB\,i\,i\pl1 \rangle
|
|
\to 0$.
|
|
|
|
The factorization properties of the amplitude also follow directly
|
|
as a consequence of positivity. For instance, let us consider the
|
|
boundary of the tree amplituhedron where the $k$ plane $(Y_1 \dots
|
|
Y_k)$ is on the plane $(Z_i Z_{i+1} Z_j Z_{j+1})$. We can e.g.
|
|
assume that $Y_1$ is a linear combination of $(Z_i, Z_{i+1}, Z_j,
|
|
Z_{j+1})$, and thus that the top row of the $C$ matrix is only
|
|
non-zero in these columns. But then, positivity remarkably forces
|
|
the $C$ matrix to ``factorize" in the form
|
|
$$
|
|
\includegraphics[scale=1]{matrix.pdf}
|
|
$$
|
|
for all possible $k_L, k_R$ such that $k_L + k_R = k - 1$. This
|
|
factorized form of the $C$ matrix in turn implies that on this
|
|
boundary, the amplituhedron does ``split" into lower-dimensional
|
|
amplituhedra in exactly the way required for the factorization of
|
|
the amplitude.
|
|
|
|
We can similarly understand the single-cut of the loop integrand.
|
|
Consider for concreteness the simplest case of the $n$ particle
|
|
one-loop MHV amplitude. On the boundary where $\langle AB\,n1
|
|
\rangle \to 0$, the $D$ matrix has the form
|
|
|
|
$$\bordermatrix{\text{}&1&2& \dots &n\cr
|
|
&1&0& \dots & -x_n\cr
|
|
&y_1&y_2&\dots&y_n \cr}$$
|
|
|
|
The connection of this $D$ matrix to the forward limit
|
|
\cite{CaronHuot:2010zt} of the NMHV tree amplitude is simple. In the
|
|
language of \cite{N5}, the forward limit in momentum-twistor space
|
|
is represented as
|
|
$$
|
|
\includegraphics[scale=.55]{pix25.pdf}
|
|
$$
|
|
we start with the tree NMHV amplitude, associated with the positive
|
|
$1 \times n$ matrix \be (y_A \, y_B \, y_1 \, y_2 \, \dots \, y_n)
|
|
\ee and first we ``add" particle $n+1$ between $n$ and $A$, which
|
|
adds three degrees of freedom $x_n, x_A, \alpha$
|
|
$$\bordermatrix{\text{}&A&B&1&2& \dots &n & n+1\cr
|
|
&x_A& \alpha x_A & 0 & 0 & \dots & -x_n & -1\cr
|
|
&y_A&y_B + \alpha y_A& y_1 & y_2 & \dots&y_n & 0\cr}$$
|
|
and we finally ``merge" $n+1,1$, which means shifting column 1 as
|
|
$c_1 \to c_ 1 - c_{n+1}$ and removing column $(n+1)$. This gives us
|
|
the matrix
|
|
|
|
$$\bordermatrix{\text{}&A&B&1&2& \dots &n\cr
|
|
&x_A& \alpha x_A & 1 & 0 & \dots & -x_n \cr
|
|
&y_A&y_B + \alpha y_A& y_1 & y_2 & \dots&y_n\cr}$$
|
|
note that the the $A,B$ columns have precisely four degrees of
|
|
freedom $x_A,\alpha, y_A, y_B$ which we can remove by $GL(2)$ acting
|
|
on the $A,B$ columns. Chopping off $A,B$ we are then left precisely
|
|
with the $D$ matrix on the single cut. This shows that the single cut of the loop integrand is the forward limit
|
|
of the tree amplitude, exactly as required by unitarity.
|
|
|
|
\section{Four Particles at All Loops}
|
|
|
|
Let us briefly describe the simplest example illustrating the novelties of positivity at
|
|
loop level, for four-particle scattering at $L$
|
|
loops. We can parametrize each $D_{(i)}$ as \be D_{(i)} =
|
|
\left(\begin{array}{cccc} 1 & x_i & 0 & -w_i \\ 0 & y_i & 1 & z_i
|
|
\end{array} \right) \ee In this simple case the positivity
|
|
constraints are just that all the $2 \times 2$ minors of $D_{(i)}$
|
|
and the $4 \times 4$ minors \be {\rm det} \left(\begin{array}{ccc} &
|
|
D_{(i)} & \\ \hdashline & D_{(j)} & \end{array} \right) \ee are
|
|
positive. This translates to \be x_i, y_i, z_i, w_i > 0, \quad (x_i
|
|
- x_j)(z_i - z_j) + (y_i - y_j)(w_i - w_j) < 0 \ee We can rephrase
|
|
this problem in a simple, purely geometrical way by defining
|
|
two dimensional vectors $\vec{a}_i = (x_i,y_i), \vec{b}_i = (z_i,
|
|
w_i)$. The points are in the upper quadrant of the plane. The mutual
|
|
positivity condition is just $(\vec{a}_i - \vec{a}_j) \cdot
|
|
(\vec{b}_i - \vec{b}_j) < 0$. Geometrically this just means that the
|
|
$\vec{a}, \vec{b}$ must be arranged so that for every pair $i,j$,
|
|
the line directed from $\vec{a}_i \to \vec{a}_j$ is pointed in the
|
|
opposite direction as the one directed from $\vec{b}_i \to
|
|
\vec{b}_j$. An example of an allowed configuration of such points
|
|
for $L=3$ is
|
|
$$
|
|
\includegraphics[scale=.6]{pix26.pdf}
|
|
$$
|
|
Finding a cell decomposition of this $4L$ dimensional space directly
|
|
gives us the integrand for the four-particle amplitude at $L$-loops.
|
|
|
|
Now, we know that the final form can be expressed as a sum over
|
|
local, planar diagrams. This makes it all the more remarkable that
|
|
no-where in the definition of our geometry problem do we
|
|
reference to diagrams of any sort, planar or not!
|
|
Nonetheless, this property is one of many that emerges from
|
|
positivity.
|
|
|
|
As we will describe at greater length in \cite{Into}, it is easy to
|
|
find a cell decomposition for the full space ``manually" at low-loop
|
|
orders. We suspect there is a more systematic approach to
|
|
understanding the geometry that might crack the problem at all loop
|
|
order. As an interesting warmup to the full problem, we can
|
|
investigate lower-dimensional ``faces" of the four-particle amplituhedron. Cellulations
|
|
of these faces corresponds to computing certain
|
|
cuts of the integrand, at all loop orders. We will discuss many of
|
|
these faces and cuts systematically in \cite{Into}. Here we will
|
|
content ourselves by presenting some especially simple but not
|
|
completely trivial examples.
|
|
|
|
Let us start by considering an extremely simple boundary of the space,
|
|
where all the $w_i \to 0$. This corresponds to having all the lines
|
|
intersect $(Z_1 Z_2)$. The positivity conditions then simply become
|
|
\be (x_i - x_j)(z_i - z_j) < 0 \ee which is trivial to
|
|
triangulate. Whatever configuration of $x$'s we have are ordered in
|
|
some way, say $x_1 < \dots < x_L$. Then we must have $z_1 > \dots
|
|
> z_L$. The $y_i$ just have to be positive. The associated form is
|
|
then trivially (we omit the measure $\prod_i dx_i dz_i d y_i$): \be
|
|
\frac{1}{y_1} \dots \frac{1}{y_L} \frac{1}{x_1} \frac{1}{x_2 - x_1}
|
|
\dots \frac{1}{x_L - x_{L-1}} \frac{1}{z_L} \frac{1}{z_{L-1} - z_L}
|
|
\dots \frac{1}{z_1 - z_2} + {\rm perm.} \ee Now, this cut is
|
|
particularly simple to understand from the point of view of the
|
|
familiar ``local" expansions of the integrand--there is only only
|
|
local diagram that can possibly contribute to this cut: the
|
|
``ladder" diagram. The corresponding cut is precisely what we
|
|
have above from positivity.
|
|
$$
|
|
\includegraphics[scale=.8]{pix19.pdf}
|
|
$$
|
|
|
|
We can continue along these lines to explore faces of the
|
|
amplituhedron which determine cuts to all loop orders that are
|
|
difficult (if not impossible) to derive in any other way. For
|
|
instance, suppose that some of the lines intersect $(Z_1
|
|
Z_2)$, so that the $w_i \to 0$ for $i = 1, \dots, L_1$ and others
|
|
intersects $(Z_3 Z_4)$, so that $y_I \to 0$ for $I =L_1+1, \dots,
|
|
L$. To pick a concrete interesting example, let choose $L-2$ lines
|
|
to intersect $(12)$ and 2 lines to intersect $(34)$. We can further
|
|
specialize the geometry and take more cuts by making the $L$'th line
|
|
pass through the point 3 -- this corresponds to sending $z_{L} \to
|
|
0$. Let us also take the $(L-1)$'st line to pass through the point 4
|
|
-- this corresponds to sending $z_{L-1}, w_{L-1} \to \infty$ with
|
|
$w_{L-1}/z_{L-1} \equiv W_{L-1}$ fixed.
|
|
|
|
We can again label the $x_i; x_I$ so they are in increasing order;
|
|
then the positivity conditions become \be x_1 < \dots < x_{L-2}, z_1
|
|
> \dots > z_{L-2}; \, x_{L-1} < x_{L} \ee and \be W_{L-1} y_i >
|
|
(x_{L-1} - x_i), \, \, w_L y_i > z_i (x_i - x_L) \ee This space is
|
|
also trivial to triangulate, but the corresponding form is more
|
|
interesting. The ordering for the $z$'s is associated with the form
|
|
$$
|
|
\frac{1}{z_{L\mi2}(z_{L\mi3}-z_{L\mi2})(z_{L\mi4}-z_{L\mi3})\dots
|
|
(z_1-z_2)}
|
|
$$
|
|
The interesting part of the space involves $x_i,y_i$. Note that if $x_i
|
|
< x_{L-1}$, the second inequality on $y_i$ is trivially satisfied
|
|
for positive $y_i$, and the only constraint on $y_i$ is just $y_i >
|
|
(x_{L-1} - x_i)/W_{L-1}$. If $x_{L-1} < x_i < x_{L}$, then both
|
|
inequalities are satisfied and we just have $y_i > 0$. Finally if
|
|
$x_i > x_L$, the first inequality is trivially satisfied and we just
|
|
have $y_i > z_i (x_i - x_L)/w_L$. Thus, given any ordering for all
|
|
the $x's$, there is an associated set of inequalities on the $y$'s,
|
|
and the corresponding form in $x,y$ space is trivially obtained. For
|
|
instance, consider the case $L=5$, and an ordering for the $x$'s
|
|
where $x_1 < x_4 < x_2 < x_5 < x_3$. The corresponding form in
|
|
$(x,y)$ space is just
|
|
\begin{align}
|
|
\frac{1}{x_1 (x_4 - x_1) (x_2 - x_4) (x_5 - x_2)(x_3 - x_5)}
|
|
\frac{1}{y_1 - (x_4 - x_1)/W_4} \frac{1}{y_2} \frac{1}{y_3 - z_3(x_3
|
|
- x_5)/w_5}
|
|
\end{align}
|
|
By summing over all the possible orderings $x$'s, we get the final
|
|
form. For general $L$, we can simply express the result (again
|
|
omitting the measure) as a sum over permutations $\sigma$:
|
|
|
|
\begin{align}
|
|
&\prod_{l=1}^{L-2}\frac{1}{(z_l -
|
|
z_{l+1})}\quad\times\hspace{-0.7cm}\sum_{\sigma; \sigma_1 < \dots <
|
|
\sigma_{L-2}; \sigma_{L-1} <
|
|
\sigma_L} \frac{1}{w_L W_{L-1}} \prod_{l=1}^{L} \frac{1}{(x_{\sigma^{-1}_l} - x_{\sigma^{-1}_{l-1}})}\\
|
|
&\hspace{4cm}\times\prod_{i=1}^{L-2} \left\{ \begin{array}{cc}
|
|
(y_i - (x_{L-1} - x_i)/W_{L-1})^{-1} & \sigma_i < \sigma_{L-1} \\
|
|
y_i^{-1} & \sigma_{L-1} < \sigma_i < \sigma_{L} \\ (y_i - (x_i -
|
|
x_L) z_i/w_L)^{-1} & \sigma_L < \sigma_i \end{array}\right\}
|
|
\nonumber
|
|
\end{align}
|
|
where we define for convenience $z_{L-1} = x_{\sigma^{-1}_0} = 0$.
|
|
|
|
This gives us non-trivial all-loop order information about
|
|
the four-particle integrand. The expression has a feature familiar
|
|
from BCFW recursion relation expressions for tree and loop level
|
|
amplitudes. Each term has certain ``spurious" poles, which cancel in
|
|
the sum. This result can be checked against the cuts of the
|
|
corresponding amplitudes that are available in ``local form". The
|
|
diagrams that contribute are of the type
|
|
$$
|
|
\includegraphics[scale=.7]{pix20.pdf}
|
|
$$
|
|
but now there are non-trivial numerator factors that don't trivially
|
|
follow from the structure of propagators. The full integrand is
|
|
available through to seven loops in the literature
|
|
\cite{Bern:2005iz,Bern:2006ew,Bern:2007ct,Bourjaily:2011hi,Eden:2012tu}.
|
|
The inspection of the available local expansions on this cut does
|
|
not indicate an obvious all-loop generalization, nor does it betray
|
|
any hint that that the final result can be expressed in the one-line
|
|
form given above. For instance just at 5 loops, the local form of
|
|
the cut is given as a sum over diagrams,
|
|
$$
|
|
\includegraphics[scale=.8]{pix29.pdf}
|
|
$$
|
|
with intricate numerator factors. If all terms are combined with a
|
|
common denominator of all physical propagators, the numerator has
|
|
347 terms. Needless to say, the complicated expression obtained in
|
|
this way perfectly matches the amplituhedron computation of the cut.
|
|
|
|
|
|
\section{Master Amplituhedron}
|
|
We have defined the amplituhedron ${\cal A}_{n,k,L}$ separately for
|
|
every $n,k$ and loop order $L$. However, a trivial feature of the
|
|
geometry is that ${\cal A}_{n,k,L}$ is contained in the ``faces" of
|
|
${\cal A}_{n^\prime, k^\prime, L^\prime}$, for $n^\prime >n,
|
|
k^\prime > k, L^\prime > L$. The objects needed to compute
|
|
scattering amplitudes for any number of particles to all loop orders
|
|
are thus contained in a ``master amplituhedron" with $n, k , L \to
|
|
\infty$.
|
|
|
|
In this vein it may also be worth considering natural
|
|
mathematical generalizations of the amplituhedron. We have already seen
|
|
that the generalized tree amplituhedron ${\cal A}_{n,k,m}$ lives in $G(k,k+m)$ and can be
|
|
defined for any even $m$. It is obvious that the amplituhedron with
|
|
$m=4$, of relevance to physics, is contained amongst the faces of
|
|
the object defined for higher $m$.
|
|
|
|
If we consider general even $m$, we can also generalize the notion of
|
|
``hiding particles" in an obvious way: adjacent particles can be hidden
|
|
in even numbers. This leads us to a bigger space in which
|
|
to embed the generalized loop amplituhedron. Instead of just considering
|
|
$G(k,k+4;L)$ of ($k-$ planes) $Y$ together with $L$ ($2-$planes) in
|
|
$m=4$ dimensional complement of $Y$, we can consider a space
|
|
$G(k,k+m;L_2,L_4, \dots, L_{m-2})$, of $k$-planes $Y$ in $(k+m)$
|
|
dimensions, together with $L_2$ (2-planes), $L_4$ (4-planes), $\dots
|
|
L_{m-2}$ ($(m-2)$-planes) in the $m$ dimensional complement of $Y$,
|
|
schematically:
|
|
$$
|
|
\includegraphics[scale=.6]{pix10.pdf}
|
|
$$
|
|
Once again we have ${\cal Y} = {\cal C} \cdot Z$, with the obvious
|
|
extension of the loop positivity conditions on ${\cal C}$ to
|
|
$G(k,n;L_2, L_4, \dots, L_{m-2})$. We can call this space ${\cal
|
|
A}_{n,k;m,L_2, \dots, L_{m-2}}(Z)$. The $m=4$ amplituhedron is again
|
|
just a particular face of this object. It would be interesting to
|
|
see whether this larger space has any interesting role to play in
|
|
understanding the $m=4$ geometry relevant to physics.
|
|
|
|
\newpage
|
|
|
|
\section{Outlook}
|
|
|
|
This paper has concerned itself with perturbative scattering
|
|
amplitudes in gauge theories. However the deeper motivations for
|
|
studying this physics, articulated in \cite{N1,N2} have to do with
|
|
some fundamental challenges of quantum gravity. We have long known
|
|
that quantum mechanics and gravity together make it impossible to
|
|
have local observables. Quantum mechanics forces us to divide the
|
|
world in two pieces--an infinite measuring apparatus and a finite
|
|
system being observed. However for any observations made in a finite
|
|
region of space-time, gravity makes it impossible to make the
|
|
apparatus arbitarily large, since it also becomes heavier, and
|
|
collapses the observation region into a black hole. In some cases
|
|
like asymptotically AdS or flat spaces, we still have precise
|
|
quantum mechanical observables, that can be measured by infinitely
|
|
large apparatuses pushed to the boundaries of space-time: boundary
|
|
correlators for AdS space and the S-matrix for flat space. The fact
|
|
that no precise observables can be associated with the inside of the
|
|
space-time strongly suggests that there should be a way of computing
|
|
these boundary observables without any reference to the interior
|
|
space-time at all. For asymptotically AdS spaces, gauge-gravity
|
|
duality \cite{Maldacena:1997re} gives us a wonderful description of
|
|
the boundary correlators of this kind, and gives a first working
|
|
example of emergent space and gravity. However, this duality is
|
|
still an equivalence between ordinary physical systems described in
|
|
standard physical language, with time running from infinite past to
|
|
infinite future. This makes the duality inapplicable to our universe
|
|
for cosmological questions. Heading back to the early universe, an
|
|
understanding of emergent time is likely necessary to make sense of
|
|
the big-bang singularity. More disturbingly, even at late times, due
|
|
to the accelerated expansion of our universe, we only have access
|
|
to a finite number of degrees of freedom, and thus the division of
|
|
the world into ``infinite" and ``finite" systems, required by
|
|
quantum mechanics to talk about precise observables, seems to be
|
|
impossible \cite{Witten:2001kn}. This perhaps indicates the need for
|
|
an extension of quantum mechanics to deal with subtle cosmological questions.
|
|
|
|
Understanding emergent space-time or possible cosmological
|
|
extensions of quantum mechanics will obviously be a tall order. The
|
|
most obvious avenue for progress is directly attacking the
|
|
quantum-gravitational questions where the relevant issues must be
|
|
confronted. But there is another strategy that takes some
|
|
inspiration from the similarly radical step taken in the transition
|
|
from classical to quantum mechanics, where classical determinism was
|
|
lost. There is a powerful clue to the coming of quantum mechanics
|
|
hidden in the structure of classical mechanics itself. While
|
|
Newton's laws are manifestly deterministic, there is a completely
|
|
different formulation of classical mechanics--in terms of the
|
|
principle of least action--which is not manifestly deterministic.
|
|
The existence of these very different starting points leading to the
|
|
same physics was somewhat mysterious to classical physicists, but
|
|
today we know why the least action formulation exists: the world is
|
|
quantum-mechanical and not deterministic, and for this reason, the
|
|
classical limit of quantum mechanics can't immediately land on
|
|
Newton's laws, but must match to some formulation of classical
|
|
physics where determinism is not a central but derived notion. The
|
|
least action principle formulation is thus much closer to quantum
|
|
mechanics than Newton's laws, and gives a better jumping off point
|
|
for making the transition to quantum mechanics as a natural
|
|
deformation, via the path integral.
|
|
|
|
We may be in a similar situation today. If there is a more
|
|
fundamental description of physics where space-time and perhaps even
|
|
the usual formulation of quantum mechanics don't appear, then even
|
|
in the limit where non-perturbative gravitational effects can be
|
|
neglected and the physics reduces to perfectly local and unitary
|
|
quantum field theory, this description is unlikely to directly
|
|
reproduce the usual formulation of field theory, but must rather
|
|
match on to some new formulation of the physics where locality and
|
|
unitarity are derived notions. Finding such reformulations of
|
|
standard physics might then better prepare us for the transition to
|
|
the deeper underlying theory.
|
|
|
|
In this paper, we have taken a baby first step in this direction,
|
|
along the lines of the program put forward in \cite{N1,N2} and
|
|
pursued in \cite{N4,N5,N6}. We have given a formulation for planar
|
|
${\cal N} = 4$ SYM scattering amplitudes with no reference to
|
|
space-time or Hilbert space, no Hamiltonians, Lagrangians or gauge
|
|
redundancies, no path integrals or Feynman diagrams, no mention of
|
|
``cuts", ``factorization channels", or recursion relations. We have
|
|
instead presented a new geometric question, to which the scattering
|
|
amplitudes are the answer. It is remarkable that such a simple
|
|
picture, merely moving from ``triangles" to ``polygons", suitably
|
|
generalized to the Grassmannian, and with an extended notion of
|
|
positivity reflecting ``hiding" particles, leads us to the
|
|
amplituhedron ${\cal A}_{n,kL}$, whose ``volume" gives us the
|
|
scattering amplitudes for a non-trivial interacting quantum field
|
|
theory in four dimensions. It is also fascinating that while in the
|
|
usual formulation of field theory, locality and unitarity are in
|
|
tension with each other, necessitating the introduction of the
|
|
familiar redundancies to accommodate both, in the new picture they
|
|
emerge together from positive geometry.
|
|
|
|
A great deal remains to be done both to establish and more fully
|
|
understand our conjecture. The usual positive Grassmannian has a
|
|
very rich cell structure. The task of understanding all possible
|
|
ways to make ordered $k \times k$ minors of a $k \times n$ matrix
|
|
positive seems daunting at first, but the key is to realize that the
|
|
``big" Grassmannian can be obtained by gluing together
|
|
(``amalgamating" \cite{FG}) ``little" $G(1,3)$'s and $G(2,3)$'s,
|
|
building up larger positive matrices from smaller ones \cite{N6}.
|
|
Remarkably, this extremely natural mathematical operation translates
|
|
directly to the physical picture of building on-shell diagrams from
|
|
gluing together elementary three-particle amplitudes. This story of
|
|
\cite{N6} is most naturally formulated in the original twistor space
|
|
or momentum space, while the amplituhedron picture is formulated in
|
|
momentum-twistor space. At tree-level, there is a direct connection
|
|
between the cells of $G(k,n)$ that cellulate the amplituhedron, and
|
|
those of $G(k+2,n)$, which give the corresponding on-shell diagram
|
|
interpretation of the cell \cite{N6}. In this way, the natural
|
|
operation of decomposing the amplituhedron into pieces is ultimately
|
|
turned into a vivid on-shell scattering picture in the original
|
|
space-time. Moving to loops, we don't have an analogous
|
|
understanding of all possible cells of the extended positive space
|
|
$G_+(k,n;L)$--we don't yet know how to systematically find positive
|
|
co-ordinates, how to think about boundaries and so on, though
|
|
certainly the on-shell representation of the loop integrand as
|
|
``non-reduced" diagrams in $G(k+2,n)$ \cite{N6} gives hope that the
|
|
necessary understanding can be reached. Having control of the cells
|
|
and positive co-ordinates for $G_+(k,n;L)$ will very likely be
|
|
necessary to properly understand the cellulation ${\cal A}_{n,k;L}$.
|
|
It would also clearly be very illuminating to find an analog of the
|
|
amplituhedron, built around positive external data in the original
|
|
twistor variables.This might also shed light on the connection
|
|
between these ideas and Witten's twistor-string theory
|
|
\cite{Witten:2003nn,Roiban:2004ix}, along the lines of
|
|
\cite{Dolan:2009wf, Nandan:2009cc, ArkaniHamed:2009dg,
|
|
Bourjaily:2010kw}.
|
|
|
|
While cell decompositions of the amplituhedron are geometrically
|
|
interesting in their own right, from the point of view of physics,
|
|
we need them only as a stepping-stone to determining the form
|
|
$\Omega_{n,k,L}$. This form was motivated by the idea of the area of
|
|
a (dual) polygon. For polygons, we have another definition of
|
|
``area", as an integral, and this gives us a completely invariant
|
|
definition for $\Omega$ free of the need for any triangulation. We
|
|
do not yet have an analog of the notion of ``dual amplituhedron",
|
|
and also no integral representation for $\Omega_{n,k,L}$. However
|
|
in \cite{Threeviews}, we will give strong circumstantial evidence
|
|
that such such an expression should exist. On a related note, while
|
|
we have a simple geometric picture for the loop integrand at any
|
|
fixed loop order, we still don't have a non-perturbative question to
|
|
which the full amplitude (rather than just the fixed-order loop
|
|
integrand) is the answer.
|
|
|
|
Note that the form $\Omega_{n,k,L}$ is given directly by
|
|
construction as a sum of ``dlog" pieces. This is a highly
|
|
non-trivial property of the integrand, made manifest (albeit less
|
|
directly) in the on-shell diagram representation of the amplitude
|
|
\cite{N6} (see also \cite{Lipstein:2012vs, Lipstein:2013xra}).
|
|
Optimistically, the great simplicity of this form should
|
|
allow a new picture for carrying out the integrations and arriving
|
|
at the final amplitudes. The crucial role that positive external
|
|
data played in our story suggests that this positive structure must
|
|
be reflected in the final amplitude in an important way. The
|
|
striking appearance of ``cluster variables" for external data in
|
|
\cite{posextamp} is an example of this.
|
|
|
|
We also hope that with a complete geometric picture for the
|
|
integrand of the amplitude in hand, we are now positioned to make direct
|
|
contact with the explosion of progress in using ideas from
|
|
integrability to determine the amplitude directly
|
|
\cite{CaronHuot:2011kk,Basso:2013vsa,Basso:2013aha, Dixon:2013eka}. A particularly promising place
|
|
to start forging this connection is with the four-particle amplitude
|
|
at all loop orders. As we noted, the positive geometry problem in
|
|
this case is especially simple, while the coefficient of the log$^2$
|
|
infrared divergence of the (log of the) amplitude gives the cusp
|
|
anomalous dimension, famously determined using integrability
|
|
techniques in \cite{BS, Beisert:2006ez,Eden:2006rx}. Another natural question is how the introduction of the spectral
|
|
parameter in on-shell diagrams given in \cite{Ferro:2013dga,Ferro:2012xw} can be realized at the level of the amplituhedron.
|
|
|
|
On-shell diagrams in ${\cal N} = 4$ SYM and the positive
|
|
Grassmannian have a close analog with on-shell diagrams in ABJM
|
|
theory and the positive null Grassmannian \cite{Huang:2013owa}, so
|
|
it is natural to expect an analog of the amplituhedron for ABJM as
|
|
well. Should we expect any of the ideas in this paper to extend to
|
|
other field theories, with less or no supersymmetry, and beyond the
|
|
planar limit? As explained in \cite{N6}, the connection between
|
|
on-shell diagrams and the Grassmannian is valid for any theory in
|
|
four dimensions, reflecting only the building-up of more complicated
|
|
on-shell processes from gluing together the basic three-particle
|
|
amplitudes. The connection with the positive Grassmannian in
|
|
particular is universal for any planar theory: only the measure on
|
|
the Grassmannian determining the on-shell form differs from theory
|
|
to theory. Furthermore, on-shell BCFW representations of scattering
|
|
amplitudes are also widely available--at loop level for planar gauge
|
|
theories, and at the very least for gravitational tree amplitudes
|
|
(where there has been much recent progress from other points of view
|
|
\cite{Hodges:2012ym,Cachazo:2012da,Cachazo:2012kg,Skinner:2013xp,Cachazo:2013hca,Cachazo:2013iea}).
|
|
As already mentioned, one of the crucial clues leading to the
|
|
amplituhedron was the myriad of different BCFW representation of
|
|
tree amplitudes, with equivalences guaranteed by remarkable rational
|
|
function identities relating BCFW terms. We have finally come to
|
|
understand these representations and identities as simple
|
|
reflections of amplituhedron geometry. As we move beyond planar
|
|
${\cal N} = 4$ SYM, we encounter even {\it more} identities with
|
|
this character, such as the BCJ relations \cite{Bern:2008qj,
|
|
Bern:2010ue}. Indeed even sticking to planar ${\cal N} = 4$ SYM,
|
|
such identities, of a fundamentally non-planar origin, give rise to
|
|
remarkable relations between amplitudes with different cyclic
|
|
orderings of the external data. It is hard to believe that these
|
|
on-shell objects and the identities they satisfy only have a
|
|
geometric ``triangulation" interpretation in the planar case, while
|
|
the even richer structure beyond the planar limit have no geometric
|
|
interpretation at all. This provides a strong impetus to search for
|
|
a geometry underlying more general theories.
|
|
|
|
Planar ${\cal N} = 4$ SYM amplitudes are Yangian invariant, a fact
|
|
that is invisible in the conventional field-theoretic description
|
|
in terms of amplitudes in one space or Wilson loops in the dual
|
|
space. We have become accustomed to such striking facts in string
|
|
theory, which has a rich spectrum of $U$ dualities, that are
|
|
impossible to make manifest simultaneously in conventional string
|
|
perturbation theory. Indeed the Yangian symmetry of planar ${\cal N}
|
|
= 4$ SYM is just fermionic $T$-duality \cite{Berkovits:2008ic}.
|
|
The amplituhedron has now given us a new description of planar
|
|
${\cal N} = 4$ SYM amplitudes which does not have a usual
|
|
space-time/quantum mechanical description, but {\it does} make all
|
|
the symmetries manifest. This is not a ``duality" in the usual sense, since we are not identifying an equivalence
|
|
between existing theories with familiar physical interpretations. We
|
|
are seeing something rather different: new mathematical
|
|
structures for representing the physics without reference to
|
|
standard physical ideas, but with all symmetries manifest. Might
|
|
there be an analogous story for superstring scattering amplitudes?
|
|
|
|
\newpage
|
|
|
|
{\Large \bf Acknowledgements} \vskip .1in
|
|
|
|
We thank Zvi Bern, Jake Bourjaily, Freddy Cachazo, Simon Caron-Huot,
|
|
Clifford Cheung, Pierre Deligne, Lance Dixon, James Drummond, Sasha
|
|
Goncharov, Song He, Johannes Henn, Andrew Hodges, Yu-tin Huang,
|
|
Jared Kaplan, Gregory Korchemsky, David Kosower, Bob MacPherson,
|
|
Juan Maldacena, Lionel Mason, David McGady, Jan Plefka, Alex
|
|
Postnikov, Amit Sever, Dave Skinner, Mark Spradlin, Matthias
|
|
Staudacher, Hugh Thomas, Pedro Vieira, Anastasia Volovich, Lauren
|
|
Williams and Edward Witten for valuable discussions. Our research in
|
|
this area over the past many years owes an enormous debt of
|
|
gratitude to Edward Witten, Andrew Hodges, and especially Freddy
|
|
Cachazo and Jake Bourjaily, without whom this work would not have
|
|
been possible. N. A.-H. is supported by the Department of Energy
|
|
under grant number DE-FG02-91ER40654. J.T. is supported in part by
|
|
the David and Ellen Lee Postdoctoral Scholarship and by DOE grant
|
|
DE-FG03-92-ER40701 and also by NSF grant PHY-0756966.
|
|
|
|
\vskip .2in
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|
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\end{document}
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