PPreprint typeset in JHEP style - PAPER VERSION
CALT-68-2872
The Amplituhedron
Nima Arkani-Hamed a and Jaroslav Trnka b a School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA b California Institute of Technology, Pasadena, CA 91125, USA
Abstract:
Perturbative scattering amplitudes in gauge theories have remarkablesimplicity and hidden infinite dimensional symmetries that are completely obscuredin the conventional formulation of field theory using Feynman diagrams. This sug-gests the existence of a new understanding for scattering amplitudes where localityand unitarity do not play a central role but are derived consequences from a differ-ent starting point. In this note we provide such an understanding for N = 4 SYMscattering amplitudes in the planar limit, which we identify as “the volume” of anew mathematical object–the Amplituhedron–generalizing the positive Grassman-nian. Locality and unitarity emerge hand-in-hand from positive geometry. a r X i v : . [ h e p - t h ] D ec ontents
1. Scattering Without Space-Time 22. Triangles → Positive Grassmannian 53. Polygons → (Tree) Amplituhedron A n,k ( Z )
64. Why Positivity? 85. Cell Decomposition 106. “Volume” as Canonical Form 127. The Superamplitude 138. Hiding Particles → Loop Positivity in G + ( k, n ; L ) A n,k,L ( Z ) – 1 – . Scattering Without Space-Time Scattering amplitudes in gauge theories are amongst the most fundamental observ-ables in physics. The textbook approach to computing these amplitudes in pertur-bation theory, using Feynman diagrams, makes locality and unitarity as manifestas possible, at the expense of introducing large amounts of gauge redundancy intoour description of the physics, leading to an explosion of apparent complexity forthe computation of amplitudes for all but the very simplest processes. Over the lastquarter-century it has become clear that this complexity is a defect of the Feyn-man diagram approach to this physics, and is not present in the final amplitudesthemselves, which are astonishingly simpler than indicated from the diagrammaticexpansion [1–7].This has been best understood for maximally supersymmetric gauge theories inthe planar limit. Planar N = 4 SYM has been used as a toy model for real physics inmany guises, but as toy models go, its application to scattering amplitudes is closerto the real world than any other. For instance the leading tree approximation toscattering amplitudes is identical to ordinary gluon scattering, and the most compli-cated part of loop amplitudes, involving virtual gluons, is also the same in N = 4SYM as in the real world.Planar N = 4 SYM amplitudes turn out to be especially simple and beauti-ful, enjoying the hidden symmetry of dual superconformal invariance [8, 9], associ-ated with a dual interpretation of scattering amplitudes as a supersymmetric Wilsonloop [10–12]. Dual superconformal symmetry combines with the ordinary conformalsymmetry to generate an infinite dimension “Yangian” symmetry [13]. Feynman dia-grams conceal this marvelous structure precisely as a consequence of making localityand unitarity manifest. For instance, the Yangian symmetry is obscured in eitherone of the standard physical descriptions either as a“scattering amplitude” in onespace-time or a “Wilson-loop” in its dual.This suggests that there must be a different formulation of the physics, wherelocality and unitarity do not play a central role, but emerge as derived features froma different starting point. A program to find a reformulation along these lines wasinitiated in [14,15], and in the context of a planar N = 4 SYM was pursued in [16–18],leading to a new physical and mathematical understanding of scattering amplitudes[19]. This picture builds on BCFW recursion relations for tree [6, 7] and loop [18,20] amplitudes, and represents the amplitude as a sum over basic building blocks,which can be physically described as arising from gluing together the elementarythree-particle amplitudes to build more complicated on-shell processes. These “on-shell diagrams” (which are essentially the same as the “twistor diagrams” of [16, 21,22]) are remarkably connected with “cells” of a beautiful new structure in algebraicgeometry, that has been studied by mathematicians over the past number of years,known as the positive Grassmannian [19, 23]. The on-shell building blocks can not– 2 –e associated with local space-time processes. Instead, they enjoy all the symmetriesof the theory, as made manifest by their connection with the Grassmannian–indeed,the infinite dimensional Yangian symmetry is easily seen to follow from “positive”diffeomorphisms [19].While these developments give a complete understanding for the on-shell buildingblocks of the amplitude, they do not go further to explain why the building blockshave to be combined in a particular way to determine the full amplitude itself. Indeed,the particular combination of on-shell diagrams is dictated by imposing that the finalresult is local and unitary–locality and unitarity specify the singularity structure ofthe amplitude, and this information is used to determine the full integrand. This isunsatisfying, since we want to see locality and unitarity emerge from more primitiveideas, not merely use them to obtain the amplitude.An important clue [17,19,24] pointing towards a deeper understanding is that theon-shell representation of scattering amplitudes is not unique: the recursion relationscan be solved in many different ways, and so the final amplitude can be expressed asa sum of on-shell processes in different ways as well. The on-shell diagrams satisfyremarkable identities–now most easily understood from their association with cellsof the positive Grassmannian–that can be used to establish these equivalences. Thisobservation led Hodges [24] to a remarkable observation for the simplest case of“NMHV” tree amplitudes, further developed in [25]: the amplitude can be thoughtof as the volume of a certain polytope in momentum twistor space. However therewas no a priori understanding of the origin of this polytope, and the picture resisteda direct generalization to more general trees or to loop amplitudes. Nonetheless, thepolytope idea motivated a continuing search for a geometric representation of theamplitude as “the volume” of “some canonical region” in “some space”, somehowrelated to the positive Grassmannian, with different “triangulations” of the spacecorresponding to different natural decompositions of the amplitude into buildingblocks.In this note we finally realize this picture. We will introduce a new mathematicalobject whose “volume” directly computes the scattering amplitude. We call thisobject the “Amplituhedron”, to denote its connection both to scattering amplitudesand positive geometry. The amplituhedron can be given a self-contained definition ina few lines as done below in section 9. We will motivate its definition from elementaryconsiderations, and show how scattering amplitudes are extracted from it.Everything flows from generalizing the notion of the “inside of a triangle in aplane”. The first obvious generalization is to the inside of a simplex in projectivespace, which further extends to the positive Grassmannian. The second generaliza-tion is to move from triangles to convex polygons, and then extend this into theGrassmannian. This gives us the amplituhedron for tree amplitudes, generalizingthe positive Grassmannian by extending the notion of positivity to include externalkinematical data. The full amplituhedron at all loop order further generalizes the– 3 –otion of positivity in a way motivated by the natural idea of “hiding particles”.Another familiar notion associated with triangles and polygons is their area. Thisis more naturally described in a projective way by a canonical 2-form with logarith-mic singularities on the boundaries of the polygon. This form also generalizes to thefull amplituhedron, and determines the (integrand of) the scattering amplitude. Thegeometry of the amplituhedron is completely bosonic, so the extraction of the su-peramplitude from this canonical form involves a novel treatment of supersymmetry,directly motivated by the Grassmannian structure.The connection between the amplituhedron and scattering amplitudes is a conjec-ture which has passed a large number of non-trivial checks, including an understand-ing of how locality and unitarity arise as consequences of positivity. Our purpose inthis note is to motivate and give the complete definition of the amplituhedron andits connection to the superamplitude in planar N = 4 SYM. The discussion will beotherwise telegraphic and few details or examples will be given. In two accompany-ing notes [26, 27], we will initiate a systematic exploration of various aspects of theassociated geometry and physics. A much more thorough exposition of these ideas,together with many examples worked out in detail, will be presented in [28]. Notation
The external data for massless n particle scattering amplitudes (for an excellentreview see [29]) are labeled as | λ a , ˜ λ a , ˜ η a (cid:105) for a = 1 , . . . , n . Here λ a , ˜ λ a are thespinor-helicity variables, determining null momenta p A ˙ Aa = λ Aa ˜ λ ˙ Aa . The ˜ η a are (four)grassmann variables for on-shell superspace. The component of the color-strippedsuperamplitude with weight 4( k + 2) in the ˜ η ’s is M n,k . We can write M n,k ( λ a , ˜ λ a , ˜ η a ) = δ ( (cid:80) a λ a ˜ λ a ) δ ( (cid:80) a λ a ˜ η a ) (cid:104) (cid:105) . . . (cid:104) n (cid:105) × M n,k ( z a , η a ) (1.1)where ( z a , η a ) are the (super) “momentum-twistor” variables [24], with z a = (cid:18) λ a µ a (cid:19) .The z a , η a are unconstrained, and determine the λ a , ˜ λ a as˜ λ a = (cid:104) a a (cid:105) µ a + + (cid:104) a + a (cid:105) µ a + (cid:104) a a + (cid:105) µ a (cid:104) a a (cid:105)(cid:104) a a + (cid:105) , ˜ η a = (cid:104) a a (cid:105) η a + + (cid:104) a + a (cid:105) η a + (cid:104) a a + (cid:105) η a (cid:104) a a (cid:105)(cid:104) a a + (cid:105) (1.2)where throughout this paper, the angle brackets (cid:104) . . . (cid:105) denotes totally antisymmetriccontraction with an (cid:15) tensor. M n,k is cyclically invariant. It is also invariant underthe little group action ( z a , η a ) → t a ( z a , η a ), so ( z a , η a ) can be taken to live in P | .At loop level, there is a well-defined notion of “the integrand” for scatteringamplitudes, which at L loops is a 4 L form. The loop integration variables are points in– 4 –he (dual) spacetime x µi , which in turn can be associated with L lines in momentum-twistor space that we denote as L ( i ) for i = 1 , · · · , L . The 4 L form is [30–32] M ( z a , η a ; L ( i ) ) (1.3)We can specify the line by giving two points L i ) , L i ) on it, which we can collect as L γ ( i ) for γ = 1 , L can also be thought of as a 2 plane in 4 dimensions. In previouswork, we have often referred to the two points on the line L , L as “ AB ”, and wewill use this notation here as well.Dual superconformal symmetry says that M n,k is invariant under the SL (4 | z a , η a ) as (super)linear transformations. The full symmetry ofthe theory is the Yangian of SL (4 |
2. Triangles → Positive Grassmannian
To begin with, let us start with the simplest and most familiar geometric object ofall, a triangle in two dimensions. Thinking projectively, the vertices are Z I , Z I , Z I where I = 1 , . . . ,
3. The interior of the triangle is a collection of points of the form Y I = c Z I + c Z I + c Z I (2.1)where we span over all c a with c a > c , c , c ) /GL (1),with all ratios c a /c b >
0, so that the c a are either all positive or all negative, buthere and in the generalizations that follow, we will abbreviate this by calling themall positive. Including the closure of the triangle replaces “positivity” with “non-negativity”, but we will continue to refer to this as “positivity” for brevity.One obvious generalization of the triangle is to an ( n −
1) dimensional simplexin a general projective space, a collection ( c , . . . , c n ) /GL (1), with c a >
0. The n -tuple ( c , . . . , c n ) /GL (1) specifies a line in n dimensions, or a point in P n − . We cangeneralize this to the space of k -planes in n dimensions–the Grassmannian G ( k, n )–which we can take to be a collection of n k − dimensional vectors modulo GL ( k )– 5 –ransformations, grouped into a k × n matrix C = c . . . c n /GL ( k ) (2.3)Projective space is the special case of G (1 , n ). The notion of positivity giving usthe “inside of a simplex” in projective space can be generalized to the Grassmannian.The only possible GL ( k ) invariant notion of positivity for the matrix C requires us tofix a particular ordering of the columns, and demand that all minors in this orderingare positive: (cid:104) c a . . . c a k (cid:105) > a < · · · < a k (2.4)We can also talk about the very closely related space of positive matrices M + ( k, n ),which are just k × n matrices with all positive ordered minors. The only differencewith the positive Grassmannian is that in M + ( k, n ) we are not moding out by GL ( k ).Note that while both M + ( k, n ) and G + ( k, n ) are defined with a given orderingfor the columns of the matrices, they have a natural cyclic structure. Indeed, if( c , . . . , c n ) give a positive matrix, then positivity is preserved under the (twisted)cyclic action c → c , . . . , c n → ( − k − c .
3. Polygons → (Tree) Amplituhedron A n,k ( Z ) Another natural generalization of a triangle is to a more general polygon with n vertices Z I , . . . , Z In . Once again we would like to discuss the interior of this region.However in general this is not canonically defined–if the points Z a are distributedrandomly, they don’t obviously enclose a region where all the Z a are all vertices. Onlyif the Z a form a convex polygon do we have a canonical “interior” to talk about:Now, convexity for the Z a is a special case of positivity in the sense of the positivematrices we have just defined. The points Z a form a closed polygon only if the 3 × n matrix with columns Z a has all positive (ordered) minors: (cid:104) Z a Z a Z a (cid:105) > a < a < a (3.1)– 6 –aving arranged for this, the interior of the polygon is given by points of the form Y I = c Z I + c Z I + . . . c n Z In with c a > c , . . . , c n ) ⊂ G + (1 , n ) , ( Z , . . . , Z n ) ⊂ M + (3 , n ) (3.3)If we jam them together to produce Y I = c a Z Ia (3.4)for fixed Z a , ranging over all c a gives us all the points on the inside of the polygon,living in G (1 , n points Z Ia in G (1 , m ), with I = 1 , . . . , m , which are positive (cid:104) Z a . . . Z a m (cid:105) > Y I = c a Z Ia , with c a > n ordered points on the moment curve in P m , Z a = (1 , t a , t a , . . . , t ma ), with t < t · · · < t n . From our point of view, forcing the points to lie on the momentcurve is overly restrictive; this is just one way of ensuring the positivity of the Z a .We can further generalize this structure into the Grassmannian. We take positiveexternal data as ( k + m ) dimensional vectors Z Ia for I = 1 , . . . , k + m . It is natural torestrict n ≥ ( k + m ), so that the external Z a fill out the entire ( k + m ) dimensionalspace. Consider the space of k -planes in this ( k + m ) dimensional space, Y ⊂ G ( k, k + m ), with co-ordinates Y Iα , α = 1 , . . . k, I = 1 , . . . , k + m (3.7)We then consider a subspace of G ( k, k + m ) determined by taking all possible “pos-itive” linear combinations of the external data, Y = C · Z (3.8)or more explicitly Y Iα = C αa Z Ia (3.9)– 7 –here C αa ⊂ G + ( k, n ) , Z Ia ⊂ M + ( k + m, n ) (3.10)It is trivial to see that this space is cyclically invariant if m is even: under thetwisted cyclic symmetry, Z n → ( − k + m − Z and c n → ( − k − c , and the productis invariant for even m .We call this space the generalized tree amplituhedron A n,k,m ( Z ). The polygon isthe simplest case with k = 1 , m = 2. Another special case is n = ( k + m ), where wecan use GL ( k + m ) transformations to set the external data to the identity matrix Z Ia = δ Ia . In this case 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 asthe tree amplituhedron A n,k ( Z ). The tree amplituhedron lives in a 4 k dimensionalspace and is not trivially visualizable. For k = 1, it is a polytope, with inequalitiesdetermined 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 4dimensional amplituhedron for n = 8, which turns out to be the space Y = c Z + . . . c Z for Z a positive external data in P :
4. Why Positivity?
We have motivated the structure of the amplituhedron by mimicking the geometricidea of the “inside” of a convex polygon. However there is a simpler and deeperorigin of the need for positivity. We can attempt to define Y = C · Z with no positiverestrictions on C or Z . But in general, this will not be projectively meaningful, andthis expression won’t allow us to define a region in G ( k, k + m ). The reason is thatfor 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 lefthand side, we obviously need positivity properties on both the Z ’s and the C ’s.– 8 –t is simple and instructive to see why positivity ensures that the Y = C · Z map is projectively well-defined. We will see this as a by-product of locating theco-dimension one boundaries of the generalized tree amplituhedron. Let us illustratethe idea already for the simplest case of the polygon with k = 1 , m = 2, with Y = c Z + . . . c n Z n . In order to look at the boundaries of the space, let us compute (cid:104) Y Z i Z j (cid:105) for some i, j . If as we sweep through all the allowed c ’s, (cid:104) Y Z i Z j (cid:105) changessign from being positive to negative, then somewhere (cid:104) Y Z i Z j (cid:105) → Y lies on theline ( Z i Z j ) in the interior of the space, thus ( Z i Z j ) should not be a boundary of thepolygon. On the other hand, if (cid:104) Y Z i Z j (cid:105) everywhere has a uniform sign, then ( Z i Z j )is a boundary of the polygon:Of course for the polygon it is trivial to directly see that the co-dimension oneboundaries are just the lines ( Z i Z i +1 ), but we wish to see this more algebraically, ina way that will generalize to the amplituhedron where “seeing” is harder. So, wecompute (cid:104) Y Z i Z j (cid:105) = (cid:88) a c a (cid:104) Z a Z i Z j (cid:105) (4.1)We can see why there is some hope for the positivity of this sum, since the c a > Z (cid:48) s are positive. It is however obvious that if i, j arenot consecutive, some of the terms in this sum will be positive, but some (where a isstuck between i, j ) will be negative. But precisely when i, j are consecutive, we geta manifestly positive sum: (cid:104) Y Z i Z i +1 (cid:105) = (cid:88) a c a (cid:104) Z a Z i Z i +1 (cid:105) > (cid:104) Z a Z i Z i +1 (cid:105) > a (cid:54) = i, i + 1, this is manifestly positive. Thus the boundariesare lines ( Z i Z i +1 ) as expected.This also tells us that the map Y = C · Z is projectively well-defined. Thereis no way to get Y →
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 wewe mod out by GL (1) on the c a . – 9 –e 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 (cid:104) Y Z i Z j Z k Z l (cid:105) = (cid:88) a c a (cid:104) Z a Z i Z j Z k Z l (cid:105) (4.3)Again, this is not in general positive. Only for ( i, j, k, l ) of the form ( i, i + 1 , j, j + 1),we have (cid:104) Y Z i Z i +1 Z j Z j +1 (cid:105) = (cid:88) a c a (cid:104) Z a Z i Z i +1 Z j Z j +1 (cid:105) > m , the boundaries are when Y is on the plane( Z i Z i +1 . . . Z i m/ − Z i m/ ). This again shows that the Y = C · Z is projectively well-defined. The result extends trivially to general k , provided the positivity of C isrespected. For m = 4 the boundaries are again when the k -plane ( Y · · · Y k ) is on( Z i Z i +1 Z j Z j +1 ), as follows from (cid:104) Y . . . Y k Z i Z i +1 Z j Z j +1 (cid:105) = (cid:88) a < ··· 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 strik-ing consequence of positivity. We will shortly understand that the location of theseboundaries are the “positive origin” of locality from the geometry of the amplituhe-dron.
5. Cell Decomposition
The tree amplituhedron can be thought of as the image of the top-cell of the thepositive Grassmannian G + ( k, n ) under the map Y = C · Z . Since dim G ( k, k + m ) = mk ≤ dim G ( k, n ) = k ( n − k ) for n ≥ k + m , this is in general a highly redundantmap. We can already see this in the simplest case of the polygon, which lives in2 dimensions, while the c a span an ( n −
1) dimensional space. The non-redundantmaps into G ( k, k + m ) can only come from the m × 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 thetriangles 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 canalso cover the inside of the polyon more nicely with non-overlapping triangles, givinga triangulation. Said in a heavy-handed way, we find a collection of 2 dimensionalcells of G + (1 , n ), so that their images in Y space are non-overlapping except onboundaries, and collectively cover the entire polygon. Of course these collections– 10 –f cells are not unique–there are many different triangulations of the polygon. Aparticularly simple one iswhich we can write as (cid:88) i (1 i i +
1) (5.1)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 polygongeneralizes to (cid:88) i 1) (5.2)This expression is immediately recognizable to physicists familiar with scatteringamplitudes in N = 4 SYM. If the ( i, j, k, l, m ) are interpreted as “R-invariants”, thisexpression is one of the canonical BCFW representations of the k = 1 “NMHV” treeamplitudes. In the positive Grassmannian representation for amplitudes [17, 19], R-invariants are precisely associated with the corresponding four-dimensional cells of G (1 , n ).For general k , m any m × k dimensional cell of G + ( k, n ) maps under Y = C · Z intosome region or cell in G ( k, k + m ). Said more explicitly, consider an m × k dimensionalcell Γ of the G + ( k, n ), with “positive co-ordinates” C Γ ( α Γ1 , . . . , α Γ m × k ) [19]. Putting Y = C ( α ) · Z and scanning over all positive α ’s, this carves out a region in G ( k, k + m )which is a corresponding cell Γ of the tree amplituhedron. A cell decomposition is acollection T of non-overlapping cells Γ which cover the entire amplituhedron.The case of immediate relevance for physics is m = 4. For any k , the BCFWdecomposition of tree amplitudes gives us a collection of 4 × k dimensional cells of thepositive Grassmannian. We have performed extensive checks for high k and n , thatfor positive external Z , under Y = C · 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– 11 –f cells of G + ( k, n ). Now we know what natural question this collection of cells areanswering: taken together they “cellulate” the tree amplituhedron. We will shortlysee how to directly associate the amplitude itself directly with the geometry of theamplituhedron. 6. “Volume” as Canonical Form Before discussing how to determine the (super)amplitude from the geometry, let usdefine the notion of a “volume” associated with the amplituhedron. As should bynow be expected, we will merely generalize a simple existing idea from the world oftriangles 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 arational 2-form in Y -space, which has logarithmic singularities on the boundaries ofthe triangle. This is naturally associated with positive co-ordinates for the triangle,if we expand Y = Z + α Z + α Z , then the form isΩ = dα α dα α (6.1)which can be re-written more invariantly asΩ = (cid:104) Y dY dY (cid:105)(cid:104) (cid:105) (cid:104) Y (cid:105)(cid:104) Y (cid:105)(cid:104) Y (cid:105) (6.2)We can extend this to a form Ω P for the convex polygon P . The defining propertyof Ω P is thatΩ P has logarithmic singularities on all the boundaries of P .Ω P can be obtained by first triangulating the polygon in some way, then summingthe elementary two-form for each triangle, for instance asΩ P = (cid:88) i Ω i i + . (6.3)Each term in this sum has singularities corresponding to Y hitting the boundaries ofthe corresponding triangle. Most of these singularities, associated with the internaledges of the triangulation, are spurious, and cancel in the sum. Of course the fullform Ω P is independent of the particular triangulation.This form is closely related to an area, not directly of the polygon P , but itsdual ˜ P , which is also a convex polygon [25]. If we dualize so that points are mappedto lines and lines to points, then a point Y inside P is mapped to a line Y outside ˜ P .If we write Ω P = (cid:104) Y d Y (cid:105) V ( Y ), then V ( Y ) is the area of ˜ P living in the euclideanspace defined by Y as the line at infinity.This form can be generalized to the tree amplituhedron in an obvious way. Wedefine a rational form Ω n,k ( Y ; Z ) with the property that– 12 – n,k ( Y ; Z ) has logarithmic singularities on all the boundaries of A n,k ( Z ).Just as for the polygon, one concrete way of computing Ω is to give a cell decom-position of the amplituhedron. For any cell Γ associated with positive co-ordinate( α Γ1 , . . . , α Γ4 k ), there is an associated form with logarithmic singularities on the bound-aries of the cell Ω Γ n,k ( Y ; Z ) = k (cid:89) i =1 dα Γ i α Γ i (6.4)For instance, consider 4 dimensional cells for k = 1, associated with cells in G + (1 , n )which are vanishing for all but columns a , . . . , a , with positive co-ordinates( α a , . . . , α a , α a = 1). Its image in Y space is simply Y = α a Z a + . . . α a Z a + Z a (6.5)and the form is dα a α a . . . dα a α a = (cid:104) Y d Y (cid:105)(cid:104) Z a Z a Z a Z a Z a (cid:105) (cid:104) Y Z a Z a Z a Z a (cid:105) . . . (cid:104) Y Z a Z a Z a Z a (cid:105) (6.6)Now, given a collection of cells T that cover the full amplituhedron, Ω n,k ( Y ; Z ) isgiven by Ω n,k ( Y ; Z ) = (cid:88) Γ ⊂ T Ω Γ n,k ( Y ; Z ) (6.7)As with the polygon, the form is independent of the particular cell decomposition.Note that the definition of the amplituhedron itself crucially depends on thepositivity of the external data Z , and this geometry in turn determines the form Ω.However, once this form is in hand, it can be analytically continued to any (complex!) Y and Z . 7. The Superamplitude We have already defined central objects in our story: the tree amplituhedron, to-gether with the associated form Ω that is loosely speaking its “volume”. The treesuper-amplitude M n,k can be directly extracted from Ω n,k ( Z ). To see how, note thatwe we can always use a GL (4 + k ) transformation to send Y → Y as Y = (cid:32) × k k × k (cid:33) (7.1)We can think of the 4 dimensional space complementary to Y , acted on by anunbroken GL (4) symmetry, as the usual P of momentum-twistor space. Accordingly,– 13 –e identify the top four components of the Z a with the usual bosonic momentum-twistor variables z a : Z a = z a ∗ ... ∗ k (7.2)We still have to decide how to interpret the remaining k entries of Z a . Clearly, if theyare 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, bysaying that some N + 1’st power of them vanishes. This is equivalent to saying thateach entry can be written as Z a = z a φ A · η A ... φ Ak · η Ak (7.3)where φ ,...,k and η a are Grassmann parameters, and A = 1 , . . . , N .Now there is a unique way to extract the amplitude. We simply localize the formΩ n,k ( Y ; Z ) to Y , and integrate over the φ ’s: M n,k ( z a , η a ) = (cid:90) d N φ . . . d N φ k (cid:90) Ω n,k ( Y ; Z ) δ k ( Y ; Y ) (7.4)Here δ k ( Y ; Y ) is a projective δ function δ k ( Y ; Y ) = (cid:90) d k × k ρ βα det( ρ ) δ k × ( k +4) ( Y Iα − ρ βα Y I β ) (7.5)Note that there is really no integral to perform in the second step; the delta functionsfully fix Y . Any form on G ( k, k + 4) is of the formΩ = (cid:104) Y . . . Y k d Y (cid:105) . . . (cid:104) Y . . . Y k d Y k (cid:105) × ω n,k ( Y ; Z ) (7.6)and our expression just says that M n,k ( z a , η a ) = (cid:90) d N φ . . . d N φ k ω n,k ( Y ; Z a ) (7.7)Note that we can define this operation for any N , however, only for N = 4 does M n,k further have weight zero under the rescaling ( z a , η a ).This connection between the form and the super-amplitude also allows us todirectly exhibit the relation between our super-amplitude expressions and the Grass-mannian formulae of [17, 19]. Consider the form in Y -space associated with a given– 14 – k dimensional cell Γ of G + ( k, n ). Then, if C Γ αa ( α , . . . , α k ) are positive co-ordinatesfor the cell, and Ω Γ = dα Γ1 α Γ1 . . . dα Γ4 k α Γ4 k is the associated form in Y space, then it is easyto show that (cid:90) d φ . . . d φ k (cid:90) Ω Γ δ k ( Y ; Y ) = (cid:90) dα Γ1 α Γ1 . . . dα Γ4 k α Γ4 k δ k | k ( C αa ( z ) Z a ) (7.8)where Z a = ( z a | η a ) are the super momentum-twistor variabes. This is precisely theformula for computing on-shell diagrams (in momentum-twistor space) as describedin [17,19,34]. Thus, while the amplituhedron geometry and the associated form Ω arepurely bosonic, we have extracted from them super-amplitudes which are manifestlysupersymmetric. Indeed, the connection to the Grassmannian shows much more–thesuperamplitude obtained for each cell is manifestly Yangian invariant [19]. 8. Hiding Particles → 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 positivitygiving us loop amplitudes.It is trivial to imagine what we might mean by hiding a single particle, but aswe will see momentarily, the idea of hiding particles is only natural if we hide pairs of adjacent particles. To pick a concrete example, suppose we have some positivematrix C with columns we’ll label ( A , B , , , . . . , m, A , B , m + 1 , . . . n ). We canalways gauge-fix the A , B and A , B columns so that the matrix takes the form A B . . . m A B m + 1 . . . n ∗ ∗ . . . ∗ ∗ . . . ∗ ∗ ∗ . . . ∗ ∗ . . . ∗ ∗ ∗ . . . ∗ ∗ . . . ∗ ∗ ∗ . . . ∗ ∗ . . . ∗ ∗ ∗ . . . ∗ ∗ . . . ∗ ... ... ... ... ... ... ... ... ... ... ...0 0 ∗ ∗ . . . ∗ ∗ . . . ∗ We would now like to “hide” the particles A , B , A , B . We do this simply bychopping out the corresponding columns. The remaining matrix can be grouped intothe form D (1) D (2) C (8.1)– 15 –ut the “hidden” particles leave an echo in the positivity properties of this matrix.The positivity of the minors involving all of ( A , B , A , B ), ( A , B ) and ( A , B )individually, as well those not involving A , B , A , B at all enforce that the orderedmaximal minors of the following matrices (cid:0) C (cid:1) , (cid:32) D (1) C (cid:33) , (cid:32) D (2) C (cid:33) , D (1) D (2) C (8.2)are all positive.We can now see why particles are most naturally hidden in pairs. If we hadinstead hidden single particles as A , A , A , . . . in separate columns, the remainingminors would be positive or negative depending on the orderings of A , A , A , . . . ,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, withpairs 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 posi-tivity 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 collectionof L D ( i ) in the ( n − k ) dimensional complement of C , schematicallyWe call this space G ( k, n ; L ), and we will denote the collection of ( D ( i ) , C ) as C .We can extend the notion of positivity to G ( k, n ; L ) by demanding that not onlythe 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 thecomplement 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 positivityby hiding particles from an underlying positive matrix, there are positive configura-tions of C that can not be obtained by hiding particles from a positive matrix in thisway. – 16 –xtending the map Y = C.Z in the obvious way to include the D ’s leads us todefine the full amplituhedron. 9. The Amplituhedron A n,k,L ( Z ) We can now give the full definition of the amplituhedron A n,k,L ( Z ). First, the ex-ternal data for n ≥ k + 4 particles is given by the vectors Z Ia living in a (4 + k )dimensional space; where a = 1 , . . . , n and I = 1 , . . . , k . The data is positive (cid:104) Z a . . . Z a k (cid:105) > a < · · · < a k (9.1)The amplituhedron lives in G ( k, k + 4; L ): the space of k planes Y in ( k + 4) di-mensions, together with L L ( i ) in the 4 dimensional complement of Y ,schematicallyWe will denote the collection of ( L ( i ) , Y ) as Y .The amplituhedron A n,k,L ( Z ) is the subspace of G ( k, k + 4; L ) consisting of all Y ’s which are a positive linear combination of the external data, Y = C · Z (9.2)More explicitly in components, the k -plane is Y Iα , and the 2-planes are L Iγ ( i ) , where γ = 1 , i = 1 , . . . , L . The amplituhedron is the space of all Y, L ( i ) of the form Y Iα = C αa Z Ia , L Iγ ( i ) = D γa ( i ) Z Ia (9.3)where as in the previous section the C αa specifies a k -plane in n -dimensions, and the D γa ( i ) are L n − k ) dimensional complement of C , with thepositivity property that for any 0 ≤ l ≤ L , all the ordered ( k + 2 l ) × ( k + 2 l ) minorsof the ( k + 2 l ) × n matrix D ( i ) ... D ( i l ) C (9.4)– 17 –re positive.The notion of cells, cell decomposition and canonical form can be extended tothe full amplituhedron. A cell Γ is associated with a set of positive co-ordinates α Γ =( α Γ1 , . . . , α Γ4( k + L ) ), rational in the C , such that for α ’s positive, C ( α ) = ( D ( i ) ( α ) , C ( α ))is in G + ( k, n ; L ). A cell decomposition is a collection T of non-intersecting cells Γwhose images under Y = C · Z cover the entire amplituhedron. The rational formΩ n,k,L ( Y ; Z ) is defined by having the property thatΩ n,k,L ( Y ; Z ) has logarithmic singularities on all the boundaries of A n,k,L ( Z )A concrete formula follows from a cell decomposition asΩ n,k,L ( Y ; Z ) = (cid:88) Γ ⊂ T k + L ) (cid:89) i =1 dα Γ i α Γ i (9.5)Of course any cell decomposition gives the same form Ω n,k,L . 10. The Loop Amplitude Form We can extract the 4 L -form for the loop integrand from Ω n,k,L in the obvious way.The 2-planes L ( i ) , being in the complement of Y , can be taken to be non-vanishingin the first 4 entries L I ( i ) = ( L ( i )2 × | × k ). Each L γ ( i ) gives us a line ( L γ =1 L γ =2 ) ( i ) (which we have also been calling ( AB ) ( i ) ) in P . These are the momentum-twistorrepresentation of the loop integration variables. The analog of equation (7.4) for theloop integrand form is M n,k ( z a , η a ; L ( γ ( i ) ) = (cid:90) d φ . . . d φ k (cid:90) Ω n,k,L ( Y, L γ ( i ) ; Z ) δ k ( Y ; Y ) (10.1)Any form on G ( k, k + 4 k ; L ) can be written asΩ = (cid:104) Y d Y (cid:105) . . . (cid:104) Y d Y k (cid:105) L (cid:89) i =1 (cid:104) Y L i ) L i ) d L i ) (cid:105)(cid:104) Y L i ) L i ) d L i ) (cid:105)× ω n,k,L ( Y, L ( i ) )( Z )(10.2)where we denoted Y = Y . . . Y k . So we have for the integrand of the all-loop ampli-tude M n,k ( z a , η a , L γ ( i ) ) = (cid:90) d φ . . . d φ k L (cid:89) i =1 (cid:104)L i ) L i ) d L i ) (cid:105)(cid:104)L i ) L i ) d L i ) (cid:105) ω n,k ( Y , L γ ( i ) ; Z a )(10.3)Already the simplest case k = 0 of the amplituhedron is interesting at loop level. At1-loop, we have a 2-plane in 4 dimensions AB , and the D matrix is just restricted to– 18 –e in G + (2 , n ). It is easy to see that the 4 dimensional cells of G + (2 , n ) are labeledby a pair of triples [ a, b, c ; x, y, z ], where the top row of the matrix is non-zero in thecolumns ( a, b, c ) and the bottom in columns ( x, y, z ). A simple collection of these (cid:88) i 1; 1 j j + 1] (10.4)beautifully covers the amplituhedron in this case. The map into G (2 , 4) for each cellis A = Z + α i Z i + α i +1 Z i +1 , B = − Z + α j Z j + α j +1 Z j +1 (10.5)and so the form associated with the cell is dα i α i dα i +1 α i +1 α j α j dα j +1 α j +1 = (cid:104) ABd A (cid:105)(cid:104) ABd B (cid:105)(cid:104) AB (1 i i + ∩ (1 j j + (cid:105) (cid:104) AB i (cid:105)(cid:104) AB i + (cid:105)(cid:104) AB i i + (cid:105)(cid:104) AB j (cid:105)(cid:104) AB j + (cid:105)(cid:104) AB j j + (cid:105) (10.6)The form Ω gives exactly the “Kermit” expansion for the MHV integrand givenin [18], now obtained without any reference to tree amplitudes, forward limits orrecursion relations.In this simple case, direct triangulation of the space is straightforward. But wecould also have worked backwards, starting with the BCFW formula, and recognizinghow each term in the “Kermit” expansion is associated with positive co-ordinates forsome cell of the amplituhedron. We could then observe that, remarkably, these cellsare non-overlapping, and together cover the full amplituhedron.In order to illustrate more of the structure of the loop amplituhedron, includingthe interplay between the “ C ” and “ D ” matrices, let us consider the 1-loop k = 1amplitude for n = 6. There are 16 terms in the BCFW recursion, which can allbe mapped back to their Y, AB space form, and in turn associated with positiveco-ordinates in the amplituhedron. For instance, one of BCFW terms is (cid:104) Y AB (cid:105)(cid:104) Y AB (561) ∩ (2345) (cid:105) (cid:104) Y AB (123) ∩ ( Y (cid:105) (cid:104) Y (cid:105)(cid:104) Y AB (561) ∩ ( Y (cid:105)(cid:104) Y AB (561) ∩ ( Y (cid:105)(cid:104) Y AB (561) ∩ ( Y (cid:105)(cid:104) Y AB (cid:105)(cid:104) Y AB (561) ∩ ( Y ∩ ( Y AB (cid:105)(cid:104) Y AB (cid:105)(cid:104) Y AB (cid:105)(cid:104) Y AB (cid:105)(cid:104) Y AB (cid:105)(cid:104) Y AB (cid:105) While it may not be immediately apparently, this is nothing but the “dlog” canonicalform associated with the following positive co-ordinates for the ( D, C ) matrix (cid:32) DC (cid:33) = x y − w − − zw xt t + t y t t z – 19 –his exercise can be repeated with all 16 BCFW terms. The corresponding ( D, C )matrices are (cid:18) x y − t − t − w zt zt t t t t (cid:19) (cid:18) x y 00 0 − w z + t t t t t t (cid:19) (cid:18) x + t y yt − t w − z − t w t t t t t t (cid:19)(cid:18) t t + x + yw y 00 0 0 wz z + t t t t t t (cid:19) (cid:18) − x y w zt t t t + x y (cid:19) (cid:18) − x − w − − y − z 01 + t t t + xt y + wt z (cid:19)(cid:18) x y − w − − zw xt t + t y t t z (cid:19) (cid:18) t t + x + yw y zw t + z t t t t (cid:19) (cid:18) − x y w zw + t t t + xt yt z (cid:19)(cid:18) − x y w zt + w t t + xt t y z (cid:19) (cid:18) x y w z + wy + t t t t t t (cid:19) (cid:18) − x y w zw + t t t t x t y z (cid:19)(cid:18) x w − z − y − z (1 + t ) y t wt + t t + t (cid:19) (cid:18) − − x − y w zw xt t + yt t + t z (1 + t ) (cid:19)(cid:18) − − x − y w zw t t t + t z (1 + t ) (cid:19) (cid:18) − − x − y w zw t x t + yt t + t z (1 + t ) (cid:19) One can easily check that for all variables positive, the bottom row of these matricesis positive, and all the ordered 3 × Y = C · Z gives an image ofthe 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 beginby fixing positive external data ( Z , · · · , Z ). We then choose any positive matrix C at random, which gives an associated point Y inside the amplituhedron. We canask whether or not this point is contained in one of the cells, by seeing whether Y can be reproduced with positive values for all eight variables of the cell. Doing thiswe 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 cellstaken together therefore give a cellulation of the amplituhedron.Note that the form shown above, associated with a BCFW term, has somephysical poles (like (cid:104) Y AB (cid:105) ), but also many unphysical poles. The unphysicalpoles 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 ofthe amplituhedron and ( b ) these collection of cells do cover the amplituhedron.It is satisfying to have a definition of the loop amplituhedron that lives directlyin the space relevant for loop amplitudes. This is in contrast with the approach tocomputing the loop integrand using recursion relations, which ultimately traces backto higher k and n tree amplitudes. Consider the simple case of the 2-loop 4-particleamplitude. We are after a form in the space of two 2-planes ( AB ) , ( AB ) in fourdimensions. The BCFW approach begins with the k = 2 , n = 8 tree amplitudes,– 20 –nd arrives at the form we are interested in after taking two “forward limits”. Butthe amplituhedron lives directly in the ( AB ) , ( AB ) space, and we can find a celldecomposition for it directly, yielding the form without having to refer to any treeamplitudes.We have understood how to directly “cellulate” the amplituhedron in a numberof other examples, and strongly suspect that there will be a general understanding forhow to do this. The BCFW decomposition of tree amplitudes seems to be associatedwith particularly nice, canonical cellulations of the tree amplituhedron. Loop levelBCFW also gives a cell decomposition. The “direct” cellulations we have foundin many cases are however simpler, without an obvious connection to the BCFWexpansion. 11. Locality and Unitarity from Positivity Locality and unitarity are encoded in the positive geometry of the amplituhedronin a beautiful way. As is well-known, locality and unitarity are directly reflected inthe singularity structure of the integrand for scattering amplitudes. In momentum-twistor language, the only allowed singularities at tree-level should occur when (cid:104) Z i Z i +1 Z j Z j +1 (cid:105) → 0; in the loop-level integrand, we can also have poles of theform (cid:104) AB i i + (cid:105) → 0, and (cid:104) AB ( i ) AB ( j ) (cid:105) → 0. Unitarity is reflected in what happensas poles are approached, schematically we have [19]Given the connection between the form Ω n,k,L and the amplitude, it is obviousthat 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, re-markably, positivity forces these faces to be precisely where (cid:104) Y . . . Y k Z i Z i +1 Z j Z j +1 (cid:105) → 0, exactly as needed for locality. The analog statement for the full loop amplituhe-dron also obviously includes (cid:104) Y · · · Y k AB i i + (cid:105) → k plane ( Y . . . Y k ) is on the plane ( Z i Z i +1 Z j Z j +1 ). We cane.g. assume that Y is a linear combination of ( Z i , Z i +1 , Z j , Z j +1 ), and thus that thetop row of the C matrix is only non-zero in these columns. But then, positivity– 21 –emarkably forces the C matrix to “factorize” in the form i i + 1 j j + 1 ↑ k L ↓ ∗ ∗ . . . ∗ ∗ . . . . . . L . . . . . . . . . . . . R . . . ↑ k R ↓ 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” intolower-dimensional amplituhedra in exactly the way required for the factorization ofthe amplitude.We can similarly understand the single-cut of the loop integrand. Consider forconcreteness the simplest case of the n particle one-loop MHV amplitude. On theboundary where (cid:104) AB n (cid:105) → 0, the D matrix has the form (cid:18) . . . n . . . − x n y y . . . y n (cid:19) The connection of this D matrix to the forward limit [35] of the NMHV treeamplitude is simple. In the language of [18], the forward limit in momentum-twistorspace is represented aswe start with the tree NMHV amplitude, associated with the positive 1 × n matrix( y A y B y y . . . y n ) (11.1)and first we “add” particle n + 1 between n and A , which adds three degrees offreedom x n , x A , α (cid:18) A B . . . n n + 1 x A αx A . . . − x n − y A y B + αy A y y . . . y n (cid:19) – 22 –nd we finally “merge” n + 1 , 1, which means shifting column 1 as c → c − c n +1 and removing column ( n + 1). This gives us the matrix (cid:18) A B . . . nx A αx A . . . − x n y A y B + αy A y y . . . y n (cid:19) note that the the A, B columns have precisely four degrees of freedom x A , α, y A , y B which we can remove by GL (2) acting on the A, B columns. Chopping off A, B weare then left precisely with the D matrix on the single cut. This shows that thesingle cut of the loop integrand is the forward limit of the tree amplitude, exactly asrequired by unitarity. 12. Four Particles at All Loops Let us briefly describe the simplest example illustrating the novelties of positivity atloop level, for four-particle scattering at L loops. We can parametrize each D ( i ) as D ( i ) = (cid:18) x i − w i y i z i (cid:19) (12.1)In this simple case the positivity constraints are just that all the 2 × D ( i ) and the 4 × (cid:32) D ( i ) D ( j ) (cid:33) (12.2)are positive. This translates to x i , y i , z i , w i > , ( x i − x j )( z i − z j ) + ( y i − y j )( w i − w j ) < (cid:126)a i = ( x i , y i ) ,(cid:126)b i = ( z i , w i ). The points are in the upper quadrantof the plane. The mutual positivity condition is just ( (cid:126)a i − (cid:126)a j ) · ( (cid:126)b i − (cid:126)b j ) < 0. Geomet-rically this just means that the (cid:126)a,(cid:126)b must be arranged so that for every pair i, j , theline directed from (cid:126)a i → (cid:126)a j is pointed in the opposite direction as the one directedfrom (cid:126)b i → (cid:126)b j . An example of an allowed configuration of such points for L = 3 is– 23 –inding a cell decomposition of this 4 L dimensional space directly gives us the inte-grand for the four-particle amplitude at L -loops.Now, we know that the final form can be expressed as a sum over local, planardiagrams. This makes it all the more remarkable that no-where in the definitionof 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 [26], it is easy to find a cell decompo-sition for the full space “manually” at low-loop orders. We suspect there is a moresystematic approach to understanding the geometry that might crack the problemat all loop order. As an interesting warmup to the full problem, we can investigatelower-dimensional “faces” of the four-particle amplituhedron. Cellulations of thesefaces corresponds to computing certain cuts of the integrand, at all loop orders. Wewill discuss many of these faces and cuts systematically in [26]. Here we will contentourselves by presenting some especially simple but not completely trivial examples.Let us start by considering an extremely simple boundary of the space, where allthe w i → 0. This corresponds to having all the lines intersect ( Z Z ). The positivityconditions then simply become( x i − x j )( z i − z j ) < x ’s we have are ordered insome way, say x < · · · < x L . Then we must have z > · · · > z L . The y i just have tobe positive. The associated form is then trivially (we omit the measure (cid:81) i dx i dz i dy i ):1 y . . . y L x x − x . . . x L − x L − z L z L − − z L . . . z − z + perm . (12.5)Now, this cut is particularly simple to understand from the point of view of thefamiliar “local” expansions of the integrand–there is only only local diagram thatcan possibly contribute to this cut: the “ladder” diagram. The corresponding cut isprecisely what we have above from positivity.We can continue along these lines to explore faces of the amplituhedron whichdetermine cuts to all loop orders that are difficult (if not impossible) to derive inany other way. For instance, suppose that some of the lines intersect ( Z Z ), sothat the w i → i = 1 , . . . , L and others intersects ( Z Z ), so that y I → 0– 24 –or I = L + 1 , . . . , L . To pick a concrete interesting example, let choose L − L ’th line pass through the point 3 – thiscorresponds to sending z L → 0. Let us also take the ( L − z L − , w L − → ∞ with w L − /z L − ≡ W L − fixed.We can again label the x i ; x I so they are in increasing order; then the positivityconditions become x < · · · < x L − , z > · · · > z L − ; x L − < x L (12.6)and W L − y i > ( x L − − x i ) , w L y i > z i ( x i − x L ) (12.7)This space is also trivial to triangulate, but the corresponding form is more interest-ing. The ordering for the z ’s is associated with the form1 z L ( z L − z L )( z L − z L ) . . . ( z − z )The interesting part of the space involves x i , y i . Note that if x i < x L − , the secondinequality on y i is trivially satisfied for positive y i , and the only constraint on y i isjust y i > ( x L − − x i ) /W L − . If x L − < x i < x L , then both inequalities are satisfiedand we just have y i > 0. Finally if x i > x L , the first inequality is trivially satisfiedand we just have y i > z i ( x i − x L ) /w L . Thus, given any ordering for all the x (cid:48) s , thereis 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 orderingfor the x ’s where x < x < x < x < x . The corresponding form in ( x, y ) space isjust 1 x ( x − x )( x − x )( x − x )( x − x ) 1 y − ( x − x ) /W y y − z ( x − x ) /w (12.8)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 overpermutations σ : L − (cid:89) l =1 z l − z l +1 ) × (cid:88) σ ; σ < ··· <σ L − ; σ L − <σ L w L W L − L (cid:89) l =1 x σ − l − x σ − l − ) (12.9) × L − (cid:89) i =1 ( y i − ( x L − − x i ) /W L − ) − σ i < σ L − y − i σ L − < σ i < σ L ( y i − ( x i − x L ) z i /w L ) − σ L < σ i – 25 –here we define for convenience z L − = x σ − = 0.This gives us non-trivial all-loop order information about the four-particle inte-grand. The expression has a feature familiar from BCFW recursion relation expres-sions 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 corre-sponding amplitudes that are available in “local form”. The diagrams that contributeare of the typebut now there are non-trivial numerator factors that don’t trivially follow from thestructure of propagators. The full integrand is available through to seven loops inthe literature [36–40]. The inspection of the available local expansions on this cutdoes not indicate an obvious all-loop generalization, nor does it betray any hint thatthat the final result can be expressed in the one-line form given above. For instancejust at 5 loops, the local form of the cut is given as a sum over diagrams,with intricate numerator factors. If all terms are combined with a common denomi-nator of all physical propagators, the numerator has 347 terms. Needless to say, thecomplicated expression obtained in this way perfectly matches the amplituhedroncomputation of the cut. 13. Master Amplituhedron We have defined the amplituhedron A n,k,L separately for every n, k and loop order L .However, a trivial feature of the geometry is that A n,k,L is contained in the “faces”of A n (cid:48) ,k (cid:48) ,L (cid:48) , for n (cid:48) > n, k (cid:48) > k, L (cid:48) > L . The objects needed to compute scatteringamplitudes for any number of particles to all loop orders are thus contained in a“master amplituhedron” with n, k, L → ∞ .– 26 –n this vein it may also be worth considering natural mathematical generaliza-tions of the amplituhedron. We have already seen that the generalized tree ampli-tuhedron A n,k,m lives in G ( k, k + m ) and can be defined for any even m . It is obviousthat the amplituhedron with m = 4, of relevance to physics, is contained amongstthe faces of the object defined for higher m .If we consider general even m , we can also generalize the notion of “hiding parti-cles” in an obvious way: adjacent particles can be hidden in even numbers. This leadsus to a bigger space in which to embed the generalized loop amplituhedron. Instead ofjust considering G ( k, k + 4; L ) of ( k − planes) Y together with L (2 − planes) in m = 4dimensional complement of Y , we can consider a space G ( k, k + m ; L , L , . . . , L m − ),of k -planes Y in ( k + m ) dimensions, together with L (2-planes), L (4-planes), . . . L m − (( m − m dimensional complement of Y , schematically:Once again we have Y = C · Z , with the obvious extension of the loop positivity con-ditions on C to G ( k, n ; L , L , . . . , L m − ). We can call this space A n,k ; m,L ,...,L m − ( Z ).The m = 4 amplituhedron is again just a particular face of this object. It wouldbe interesting to see whether this larger space has any interesting role to play inunderstanding the m = 4 geometry relevant to physics.– 27 – 4. Outlook This paper has concerned itself with perturbative scattering amplitudes in gauge the-ories. However the deeper motivations for studying this physics, articulated in [14,15]have to do with some fundamental challenges of quantum gravity. We have longknown that quantum mechanics and gravity together make it impossible to have lo-cal observables. Quantum mechanics forces us to divide the world in two pieces–aninfinite measuring apparatus and a finite system being observed. However for anyobservations made in a finite region of space-time, gravity makes it impossible tomake the apparatus arbitarily large, since it also becomes heavier, and collapses theobservation region into a black hole. In some cases like asymptotically AdS or flatspaces, we still have precise quantum mechanical observables, that can be measuredby infinitely large apparatuses pushed to the boundaries of space-time: boundarycorrelators for AdS space and the S-matrix for flat space. The fact that no preciseobservables can be associated with the inside of the space-time strongly suggests thatthere should be a way of computing these boundary observables without any refer-ence to the interior space-time at all. For asymptotically AdS spaces, gauge-gravityduality [41] gives us a wonderful description of the boundary correlators of this kind,and gives a first working example of emergent space and gravity. However, thisduality is still an equivalence between ordinary physical systems described in stan-dard physical language, with time running from infinite past to infinite future. Thismakes the duality inapplicable to our universe for cosmological questions. Headingback to the early universe, an understanding of emergent time is likely necessary tomake sense of the big-bang singularity. More disturbingly, even at late times, dueto the accelerated expansion of our universe, we only have access to a finite numberof degrees of freedom, and thus the division of the world into “infinite” and “finite”systems, required by quantum mechanics to talk about precise observables, seemsto be impossible [42]. This perhaps indicates the need for an extension of quantummechanics to deal with subtle cosmological questions.Understanding emergent space-time or possible cosmological extensions of quan-tum mechanics will obviously be a tall order. The most obvious avenue for progressis directly attacking the quantum-gravitational questions where the relevant issuesmust be confronted. But there is another strategy that takes some inspiration fromthe similarly radical step taken in the transition from classical to quantum mechan-ics, where classical determinism was lost. There is a powerful clue to the comingof quantum mechanics hidden in the structure of classical mechanics itself. WhileNewton’s laws are manifestly deterministic, there is a completely different formu-lation of classical mechanics–in terms of the principle of least action–which is notmanifestly deterministic. The existence of these very different starting points leadingto the same physics was somewhat mysterious to classical physicists, but today weknow why the least action formulation exists: the world is quantum-mechanical and– 28 –ot deterministic, and for this reason, the classical limit of quantum mechanics can’timmediately land on Newton’s laws, but must match to some formulation of classi-cal physics where determinism is not a central but derived notion. The least actionprinciple 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 mechanicsas a natural deformation, via the path integral.We may be in a similar situation today. If there is a more fundamental de-scription of physics where space-time and perhaps even the usual formulation ofquantum mechanics don’t appear, then even in the limit where non-perturbativegravitational effects can be neglected and the physics reduces to perfectly local andunitary quantum field theory, this description is unlikely to directly reproduce theusual formulation of field theory, but must rather match on to some new formulationof the physics where locality and unitarity are derived notions. Finding such refor-mulations of standard physics might then better prepare us for the transition to thedeeper underlying theory.In this paper, we have taken a baby first step in this direction, along the lines ofthe program put forward in [14,15] and pursued in [17–19]. We have given a formula-tion for planar N = 4 SYM scattering amplitudes with no reference to space-time orHilbert space, no Hamiltonians, Lagrangians or gauge redundancies, no path integralsor Feynman diagrams, no mention of “cuts”, “factorization channels”, or recursionrelations. We have instead presented a new geometric question, to which the scatter-ing amplitudes are the answer. It is remarkable that such a simple picture, merelymoving from “triangles” to “polygons”, suitably generalized to the Grassmannian,and with an extended notion of positivity reflecting “hiding” particles, leads us tothe amplituhedron A n,kL , whose “volume” gives us the scattering amplitudes for anon-trivial interacting quantum field theory in four dimensions. It is also fascinatingthat while in the usual formulation of field theory, locality and unitarity are in ten-sion with each other, necessitating the introduction of the familiar redundancies toaccommodate both, in the new picture they emerge together from positive geometry.A great deal remains to be done both to establish and more fully understandour conjecture. The usual positive Grassmannian has a very rich cell structure. Thetask of understanding all possible ways to make ordered k × k minors of a k × n matrix positive seems daunting at first, but the key is to realize that the “big”Grassmannian can be obtained by gluing together (“amalgamating” [43]) “little” G (1 , G (2 , G ( k, n ) that cellulate the amplituhedron, and those of G ( k + 2 , n ), which– 29 –ive the corresponding on-shell diagram interpretation of the cell [19]. In this way,the natural operation of decomposing the amplituhedron into pieces is ultimatelyturned into a vivid on-shell scattering picture in the original space-time. Moving toloops, we don’t have an analogous understanding of all possible cells of the extendedpositive 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-shellrepresentation of the loop integrand as “non-reduced” diagrams in G ( k + 2 , n ) [19]gives hope that the necessary understanding can be reached. Having control of thecells and positive co-ordinates for G + ( k, n ; L ) will very likely be necessary to properlyunderstand the cellulation A n,k ; L . It would also clearly be very illuminating to findan analog of the amplituhedron, built around positive external data in the originaltwistor variables.This might also shed light on the connection between these ideasand Witten’s twistor-string theory [4, 44], along the lines of [45–48].While cell decompositions of the amplituhedron are geometrically interesting intheir own right, from the point of view of physics, we need them only as a stepping-stone to determining the form Ω n,k,L . This form was motivated by the idea of thearea of a (dual) polygon. For polygons, we have another definition of “area”, asan integral, and this gives us a completely invariant definition for Ω free of theneed for any triangulation. We do not yet have an analog of the notion of “dualamplituhedron”, and also no integral representation for Ω n,k,L . However in [27], wewill 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 integrandat any fixed loop order, we still don’t have a non-perturbative question to which thefull amplitude (rather than just the fixed-order loop integrand) is the answer.Note that the form Ω 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 (albeitless directly) in the on-shell diagram representation of the amplitude [19] (see also[49, 50]). Optimistically, the great simplicity of this form should allow a new picturefor carrying out the integrations and arriving at the final amplitudes. The crucial rolethat positive external data played in our story suggests that this positive structuremust be reflected in the final amplitude in an important way. The striking appearanceof “cluster variables” for external data in [51] is an example of this.We also hope that with a complete geometric picture for the integrand of theamplitude in hand, we are now positioned to make direct contact with the explo-sion of progress in using ideas from integrability to determine the amplitude di-rectly [52–55]. A particularly promising place to start forging this connection is withthe four-particle amplitude at all loop orders. As we noted, the positive geometryproblem in this case is especially simple, while the coefficient of the log infrared di-vergence of the (log of the) amplitude gives the cusp anomalous dimension, famouslydetermined using integrability techniques in [56–58]. Another natural question ishow the introduction of the spectral parameter in on-shell diagrams given in [59, 60]– 30 –an be realized at the level of the amplituhedron.On-shell diagrams in N = 4 SYM and the positive Grassmannian have a closeanalog with on-shell diagrams in ABJM theory and the positive null Grassman-nian [61], so it is natural to expect an analog of the amplituhedron for ABJM aswell. Should we expect any of the ideas in this paper to extend to other field the-ories, with less or no supersymmetry, and beyond the planar limit? As explainedin [19], the connection between on-shell diagrams and the Grassmannian is validfor any theory in four dimensions, reflecting only the building-up of more compli-cated on-shell processes from gluing together the basic three-particle amplitudes.The connection with the positive Grassmannian in particular is universal for anyplanar theory: only the measure on the Grassmannian determining the on-shell formdiffers from theory to theory. Furthermore, on-shell BCFW representations of scat-tering 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 muchrecent progress from other points of view [62–67]). As already mentioned, one of thecrucial clues leading to the amplituhedron was the myriad of different BCFW rep-resentation of tree amplitudes, with equivalences guaranteed by remarkable rationalfunction identities relating BCFW terms. We have finally come to understand theserepresentations and identities as simple reflections of amplituhedron geometry. Aswe move beyond planar N = 4 SYM, we encounter even more identities with thischaracter, such as the BCJ relations [68, 69]. Indeed even sticking to planar N = 4SYM, such identities, of a fundamentally non-planar origin, give rise to remarkablerelations between amplitudes with different cyclic orderings of the external data. Itis hard to believe that these on-shell objects and the identities they satisfy onlyhave a geometric “triangulation” interpretation in the planar case, while the evenricher structure beyond the planar limit have no geometric interpretation at all. Thisprovides a strong impetus to search for a geometry underlying more general theories.Planar N = 4 SYM amplitudes are Yangian invariant, a fact that is invisiblein the conventional field-theoretic description in terms of amplitudes in one spaceor Wilson loops in the dual space. We have become accustomed to such strikingfacts in string theory, which has a rich spectrum of U dualities, that are impossibleto make manifest simultaneously in conventional string perturbation theory. Indeedthe Yangian symmetry of planar N = 4 SYM is just fermionic T -duality [70]. Theamplituhedron has now given us a new description of planar N = 4 SYM amplitudeswhich does not have a usual space-time/quantum mechanical description, but does make all the symmetries manifest. This is not a “duality” in the usual sense, sincewe are not identifying an equivalence between existing theories with familiar phys-ical interpretations. We are seeing something rather different: new mathematicalstructures for representing the physics without reference to standard physical ideas,but with all symmetries manifest. Might there be an analogous story for superstringscattering amplitudes? – 31 – cknowledgements We thank Zvi Bern, Jake Bourjaily, Freddy Cachazo, Simon Caron-Huot, CliffordCheung, 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, MatthiasStaudacher, Hugh Thomas, Pedro Vieira, Anastasia Volovich, Lauren Williams andEdward Witten for valuable discussions. Our research in this area over the past manyyears owes an enormous debt of gratitude to Edward Witten, Andrew Hodges, andespecially Freddy Cachazo and Jake Bourjaily, without whom this work would nothave been possible. N. A.-H. is supported by the Department of Energy under grantnumber DE-FG02-91ER40654. J.T. is supported in part by the David and Ellen LeePostdoctoral Scholarship and by DOE grant DE-FG03-92-ER40701 and also by NSFgrant PHY-0756966. References [1] S. J. Parke and T. R. Taylor, Phys. Rev. Lett. , 2459 (1986).[2] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B , 217(1994) [hep-ph/9403226].[3] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B , 59(1995) [hep-ph/9409265].[4] E. Witten, Commun. Math. Phys. , 189 (2004) [hep-th/0312171].[5] F. Cachazo, P. Svrcek and E. Witten, JHEP , 006 (2004) [hep-th/0403047].[6] R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B , 499 (2005) [hep-th/0412308].[7] R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. , 181602 (2005)[hep-th/0501052].[8] L. F. Alday and J. M. Maldacena, JHEP , 064 (2007) [arXiv:0705.0303[hep-th]].[9] J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev, JHEP , 064(2007) [hep-th/0607160].[10] S. Caron-Huot, JHEP , 058 (2011) [arXiv:1010.1167 [hep-th]].[11] L. J. Mason and D. Skinner, JHEP , 018 (2010) [arXiv:1009.2225 [hep-th]].[12] L. F. Alday, B. Eden, G. P. Korchemsky, J. Maldacena and E. Sokatchev, JHEP , 123 (2011) [arXiv:1007.3243 [hep-th]]. – 32 – 13] J. M. Drummond, J. M. Henn and J. Plefka, JHEP , 046 (2009)[arXiv:0902.2987 [hep-th]].[14] N. Arkani-Hamed and J. Kaplan, JHEP , 076 (2008) [arXiv:0801.2385 [hep-th]].[15] N. Arkani-Hamed, F. Cachazo and J. Kaplan, JHEP , 016 (2010)[arXiv:0808.1446 [hep-th]].[16] N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, JHEP , 110 (2010)[arXiv:0903.2110 [hep-th]].[17] N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, JHEP , 020 (2010)[arXiv:0907.5418 [hep-th]].[18] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, JHEP , 041 (2011) [arXiv:1008.2958 [hep-th]].[19] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov andJ. Trnka, arXiv:1212.5605 [hep-th].[20] R. H. Boels, JHEP , 113 (2010) [arXiv:1008.3101 [hep-th]].[21] A. P. Hodges and S. Huggett, Surveys High Energ. Phys. , 333 (1980).[22] A. P. Hodges, hep-th/0503060.[23] A. Postnikov, arXiv:math/0609764.[24] A. Hodges, JHEP , 135 (2013) [arXiv:0905.1473 [hep-th]].[25] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, JHEP ,081 (2012) [arXiv:1012.6030 [hep-th]].[26] N. Arkani-Hamed and J. Trnka, “Into the Amplituhedron”, to appear.[27] N. Arkani-Hamed, A. Hodges and J. Trnka, “Three Views of the Amplituhedron”, toappear.[28] N. Arkani-Hamed and J. Trnka, “Scattering Amplitudes from Positive Geometry”, inpreparation.[29] H. Elvang and Y. -t. Huang, arXiv:1308.1697 [hep-th].[30] A. Hodges, JHEP , 051 (2013) [arXiv:1004.3323 [hep-th]].[31] L. Mason and D. Skinner, J. Phys. A , 135401 (2011) [arXiv:1004.3498 [hep-th]].[32] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, JHEP , 125 (2012)[arXiv:1012.6032 [hep-th]].[33] http://en.wikipedia.org/wiki/Cyclic polytope – 33 – 34] L. J. Mason and D. Skinner, JHEP , 045 (2009) [arXiv:0909.0250 [hep-th]].[35] S. Caron-Huot, JHEP , 080 (2011) [arXiv:1007.3224 [hep-ph]].[36] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Phys. Rev. D , 085010 (2007) [hep-th/0610248].[37] Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D , 085001 (2005)[hep-th/0505205].[38] Z. Bern, J. J. M. Carrasco, H. Johansson and D. A. Kosower, Phys. Rev. D ,125020 (2007) [arXiv:0705.1864 [hep-th]].[39] J. L. Bourjaily, A. DiRe, A. Shaikh, M. Spradlin and A. Volovich, JHEP , 032(2012) [arXiv:1112.6432 [hep-th]].[40] B. Eden, P. Heslop, G. P. Korchemsky and E. Sokatchev, Nucl. Phys. B , 450(2012) [arXiv:1201.5329 [hep-th]].[41] J. M. Maldacena, [hep-th/9711200].[42] E. Witten, hep-th/0106109.[43] V. V. Fock and A. B. Goncharov, Ann. Sci. L’Ecole Norm. Sup. (2009) ,arXiv:math.AG/0311245.[44] R. Roiban, M. Spradlin and A. Volovich, Phys. Rev. Lett. , 102002 (2005)[hep-th/0412265].[45] L. Dolan and P. Goddard, JHEP , 032 (2009) [arXiv:0909.0499 [hep-th]].[46] D. Nandan, A. Volovich and C. Wen, JHEP , 061 (2010) [arXiv:0912.3705[hep-th]].[47] N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, JHEP , 049 (2011)[arXiv:0912.4912 [hep-th]].[48] J. L. Bourjaily, J. Trnka, A. Volovich and C. Wen, JHEP , 038 (2011)[arXiv:1006.1899 [hep-th]].[49] A. E. Lipstein and L. Mason, JHEP , 106 (2013) [arXiv:1212.6228 [hep-th]].[50] A. E. Lipstein and L. Mason, arXiv:1307.1443 [hep-th].[51] J. Golden, A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich,arXiv:1305.1617 [hep-th].[52] S. Caron-Huot and S. He, JHEP , 174 (2012) [arXiv:1112.1060 [hep-th]].[53] B. Basso, A. Sever and P. Vieira, Phys. Rev. Lett. , 091602 (2013)[arXiv:1303.1396 [hep-th]]. – 34 – 54] B. Basso, A. Sever and P. Vieira, arXiv:1306.2058 [hep-th].[55] L. J. Dixon, J. M. Drummond, M. von Hippel and J. Pennington, arXiv:1308.2276[hep-th].[56] N. Beisert and M. Staudacher, Nucl. Phys. B , 439 (2003) [hep-th/0307042].[57] N. Beisert, B. Eden and M. Staudacher, J. Stat. Mech. , P01021 (2007)[hep-th/0610251].[58] B. Eden and M. Staudacher, J. Stat. Mech. , P11014 (2006) [hep-th/0603157].[59] L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, arXiv:1308.3494[hep-th].[60] L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Phys. Rev. Lett. , no. 12, 121602 (2013) [arXiv:1212.0850 [hep-th]].[61] Y. -t. Huang and C. Wen, arXiv:1309.3252 [hep-th].[62] A. Hodges, arXiv:1204.1930 [hep-th].[63] F. Cachazo and Y. Geyer, arXiv:1206.6511 [hep-th].[64] F. Cachazo and D. Skinner, Phys. Rev. Lett. , 161301 (2013) [arXiv:1207.0741[hep-th]].[65] D. Skinner, arXiv:1301.0868 [hep-th].[66] F. Cachazo, S. He and E. Y. Yuan, arXiv:1307.2199 [hep-th].[67] F. Cachazo, S. He and E. Y. Yuan, arXiv:1309.0885 [hep-th].[68] Z. Bern, J. J. M. Carrasco and H. Johansson, Phys. Rev. D , 085011 (2008)[arXiv:0805.3993 [hep-ph]].[69] Z. Bern, J. J. M. Carrasco and H. Johansson, [arXiv:1004.0476 [hep-th]].[70] N. Berkovits and J. Maldacena, JHEP , 062 (2008) [arXiv:0807.3196 [hep-th]]., 062 (2008) [arXiv:0807.3196 [hep-th]].