JHEP10(2014)030 Springer October 6, 2014 : September 4, 2014 September 8, 2014 : : 10.1007/JHEP10(2014)030 Published Received Accepted doi: Published for SISSA by b = 4 SYM scattering amplitudes in the planar limit, N [email protected] , and Jaroslav Trnka . 3 a 1312.2007 The Authors. c Supersymmetric gauge theory, Scattering Amplitudes

Perturbative scattering amplitudes in gauge theories have remarkable simplic- , [email protected] 1 Einstein Dr., Princeton, NJWalter 08540, Burke U.S.A. Institute for1200 Theoretical Physics, E California Colorado Blvd., Institute Pasadena, of CA Technology, 91125,E-mail: U.S.A. School of Natural Sciences, Institute for Advanced Study, b a Open Access Article funded by SCOAP ArXiv ePrint: provide such an understanding for which we identify as “thegeneralizing volume” the of positive a Grassmannian. new mathematicalpositive Locality object and geometry. — unitarity the emerge Amplituhedron hand-in-hand — from Keywords: Abstract: ity and hidden infiniteventional dimensional formulation of symmetries field that theory areof using a completely Feynman diagrams. new obscured understanding This in for suggestsa the scattering the central con- existence amplitudes role where but are locality derived and consequences unitarity from do a not different play starting point. In this note we Nima Arkani-Hamed The Amplituhedron JHEP10(2014)030 13 ) L ; 5 ) k, n ( Z ( + ]. 7 G 19 – n,k 1 4 A – 1 – 15 1 ) 11 Z ( 21 16 24 n,k,L A 12 loop positivity in 9 = 4 SYM has been used as a toy model for real physics in many → 8 N positive Grassmannian (tree) Amplituhedron → → 25 This has been best understood for maximally supersymmetric gauge theories in the that this complexity isis a not defect present of inindicated the from the the Feynman final diagrammatic diagram amplitudes expansion approach [ themselves, to which this areplanar physics, astonishingly limit. and simpler Planar than Scattering amplitudes in gauge theoriesphysics. are amongst The the textbook mostusing approach fundamental Feynman observables to in diagrams, computingexpense these makes amplitudes of locality in introducing and perturbationphysics, large unitarity leading theory, amounts to as an of manifest explosionfor gauge of as all apparent redundancy but possible, complexity for the into at the very our computation simplest the of description processes. amplitudes of Over the the last quarter-century it has become clear 1 Scattering without space-time 12 Four particles at all loops 13 Master Amplituhedron 14 Outlook 9 The Amplituhedron 10 The loop amplitude form 11 Locality and unitarity from positivity 6 “Volume” as canonical form 7 The superamplitude 8 Hiding particles 3 Polygons 4 Why positivity? 5 Cell decomposition Contents 1 Scattering without space-time 2 Triangles JHEP10(2014)030 ], ] to 15 , 24 14 ]. Dual 12 – 10 that the final result is ]. This picture builds ]. The on-shell building ], leading to a new phys- the building blocks have 19 ], associated with a dual 23 18 , – 9 ] amplitudes, and represents , 19 why 8 16 imposing 20 , 18 = 4 SYM as in the real world. N ]. Feynman diagrams conceal this marvelous ] and loop [ – 2 – 13 7 , 6 ]) are remarkably connected with “cells” of a beautiful = 4 SYM was pursued in [ ] pointing towards a deeper understanding is that the on- 22 , N 24 , 21 , to determine the full integrand. This is unsatisfying, since we 19 16 , 17 used ]. 19 = 4 SYM amplitudes turn out to be especially simple and beautiful, enjoying N An important clue [ While these developments give a complete understanding for the on-shell building This suggests that there must be a different formulation of the physics, where locality Planar solved in many different ways,shell and processes so in the different final ways amplitude as— well. can now be The most expressed on-shell easily diagrams as understoodnian satisfy a — remarkable from sum that identities their of can association be on- used witha to cells remarkable establish these of observation equivalences. for the the This positive observation simplest Grassman- led case Hodges [ of “NMHV” tree amplitudes, further devel- local and unitary — locality andand unitarity specify this the information singularity is structure ofwant the amplitude, to see locality andto unitarity obtain emerge the from amplitude. more primitive ideas, not merelyshell use representation them of scattering amplitudes is not unique: the recursion relations can be diffeomorphisms [ blocks of the amplitude,to they be do combined not inparticular go combination a further of to particular on-shell explain diagrams way is to dictated determine by the full amplitude itself. Indeed, the new structure in algebraicpast geometry, number of that years, has knownblocks been as can the studied not positive by be Grassmanniansymmetries mathematicians [ associated of over the with the theory, local asindeed, space-time made the manifest processes. by infinite their dimensional Instead, connection Yangian they with symmetry enjoy the is Grassmannian all easily — seen the to follow from “positive” on BCFW recursion relationsthe for amplitude tree as [ aarising sum from over gluing basic together buildingplicated the blocks, on-shell elementary which three-particle processes. amplitudes can Thesethe to be “twistor “on-shell build physically diagrams” diagrams” more of described (which [ com- as are essentially the same as and unitarity do notstarting play point. a A central program to role,and find in but a the reformulation emerge along context as these ofical lines a derived was and planar features initiated mathematical in from [ understanding a different of scattering amplitudes [ interpretation of scattering amplitudessuperconformal symmetry as combines a with the supersymmetricinfinite ordinary Wilson dimension conformal symmetry loop “Yangian” to symmetry [ generatestructure [ precisely an as a consequence ofthe making locality Yangian symmetry and is unitarity manifest. obscured in Foras instance, either a“scattering one of amplitude” the in standard one physical space-time descriptions either or a “Wilson-loop” in its dual. world than any other. For instanceis the identical leading to tree ordinary approximation gluon to scattering,involving scattering and virtual amplitudes the gluons, most is complicated also part of the loop same amplitudes, in the hidden symmetry of dual superconformal invariance [ guises, but as toy models go, its application to scattering amplitudes is closer to the real JHEP10(2014)030 are the spinor- a ], we will initiate ˜ λ , a 27 , λ are (four) grassmann 26 a η . Here . The ˜ ˙ A a ˜ λ , . . . , n A a λ = 1 = a ˙ A A a particle scattering amplitudes (for an excel- for p i n a ˜ – 3 – η , a ˜ λ , a λ | ]. 28 = 4 SYM. The discussion will be otherwise telegraphic and N ]) are labeled as 29 The external data for massless ]: the amplitude can be thought of as the volume of a certain polytope in momen- 25 The connection between the amplituhedron and scattering amplitudes is a conjecture Another familiar notion associated with triangles and polygons is their area. This Everything flows from generalizing the notion of the “inside of a triangle in a plane”. In this note we finally realize this picture. We will introduce a new mathematical Notation. lent review see [ helicity variables, determining nullvariables momenta for on-shell superspace. The component of the color-stripped superamplitude superamplitude in planar few details or examples willa be systematic given. exploration In ofmuch two more accompanying various notes thorough aspects [ exposition ofin of detail, the these will ideas, associated be together presented geometry with in and many [ physics. examples worked out A the Grassmannian structure. which has passed a largelocality number and of unitarity non-trivial arise checks,motivate including as and an consequences give understanding the of of complete positivity. how definition Our of the purpose amplituhedron in and this its note connection is to to the is more naturally describedsingularities in on a the projectiveamplituhedron, boundaries and way determines of by the (integrand the aof of) canonical polygon. the the 2-form scattering amplituhedron amplitude. is with Thisthis completely The logarithmic geometry bosonic, form canonical so form also the involves generalizes extraction a of to novel the the treatment superamplitude of from full supersymmetry, directly motivated by amplituhedron for tree amplitudes,the generalizing notion the of positivity positive to Grassmannianloop include by external order kinematical extending further data. generalizesidea The the full of notion amplituhedron “hiding at of particles”. all positivity in a way motivated by the natural and show how scattering amplitudes are extracted from it. The first obvious generalizationfurther extends is to to the thetriangles positive to inside Grassmannian. convex polygons, of The and a second then extend generalization simplex this is into in the to Grassmannian. projective move This from space, gives us which the corresponding to different natural decompositions of the amplitude intoobject building blocks. whose “volume” directlythe “Amplituhedron”, computes to the denote scattering itsgeometry. connection amplitude. both The to We amplituhedron scattering calldone can amplitudes this below and be object in positive given section a 9. self-contained We definition will in motivate a its few definition from lines elementary as considerations, tum twistor space. However theretope, was and no the a picture prioriplitudes. resisted understanding a of Nonetheless, the the direct polytope origin generalization idea ofresentation to motivated this of more a poly- the continuing general search amplitude trees for assomehow a or “the related geometric to to volume” rep- loop the of positive am- “some Grassmannian, canonical with region” different in “triangulations” “some of the space”, space oped in [ JHEP10(2014)030 ) i ( γ L (2.1) (1.2) . The where ! I 3 a a λ µ 4) symme- ,Z | I 2

,Z = I 1 (1.1) a ) Z z . a , 4 | 1 , η 3 1 a − P − z a ”, and we will use this a ( µ η i ], with i n,k AB 24 ] + 1 + 1 32 as “ × M – a a a a 2 h ) h 30 lines in momentum-twistor space L a I 3 on it, which we can collect as , + ˜ η i denotes totally antisymmetric con- + i 1 Z a ) L a i a i 3 L λ µ c η is invariant under the SL(4 2( + 1 i (1.3) a i + 1 as i ) 1 ... L 1 + ) 1 h , a i P n,k I ( ) n form is [ 2 a a − ( a a ˜ − i λ h (2.2) 8 L , Z ih a 1( ih a 0 M L δ ; a 2 a ) a a c L λ ... a 1 > ) can be taken to live in 1 i ˜ λ , η + 1 – 4 – + a + 1 a a a − − c I a 12 1 a z λ , η h h a h ( a Z a h a . We can write h . The 4 1 z + + c M P n,k ,L ( = +1 4 +1 a M a δ I is cyclically invariant. It is also invariant under the little ··· µ η ), so ( i i , Y a a a n,k ) = ’s is , η 1 1 a η = 1 a ˜ η form. The loop integration variables are points in the (dual) z M − − i with , ( a a a a a L t h h ˜ λ c for , a ) → = = 4). i λ positive Grassmannian | ( ) as (super)linear transformations. The full symmetry of the theory is ) ( a a a tensor. a L ˜ η ˜ λ  , η n,k can also be thought of as a 2 plane in 4 dimensions. In previous work, → , η + 2) in the ˜ a a z M k z L , which in turn can be associated with ) are the (super) “momentum-twistor” variables [ loops is a 4 µ i a 2. x 3. The interior of the triangle is a collection of points of the form L , , η a z are unconstrained, and determine the = 1 ,..., a Dual superconformal symmetry says that At loop level, there is a well-defined notion of “the integrand” for scattering amplitudes, γ , η = 1 a where we span over all 2 Triangles To begin with, leta us triangle start in withI two the dimensions. simplest and Thinking most projectively, familiar the geometric object vertices are of all, try acting on ( the Yangian of SL(4 We can specify the linefor by giving two points we have often referred tonotation the here as two points well. on the line which at spacetime that we denote as where throughout this paper,traction the with angle an brackets group action ( z where ( with weight 4( JHEP10(2014)030 ) n → 1 (2.4) c -tuple n vertices ), which , . . . , c (1), with 1 n c k, n ( /GL + ) ). 3 k 0. The M ( , c ) — which we can 2 > . We can generalize GL , c 1 1 k, n a c ( − c n G P 1) dimensional simplex in ) transformations, grouped k requires us to fix a particular ( − k are distributed randomly, they C n GL (2.3) (1), with a ) ) < a k Z ( Z ( /GL ··· ) n /GL < n,k ) are defined with a given ordering for ). The notion of positivity giving us the 1    A a , n k, n n ( (1 , . . . , c + 1 G – 5 – c G 0 for ) we are not moding out by . . . c > dimensions, or a point in 1 k, n i c n ( k a + ) and    are either all positive or all negative, but here and in the M = dimensions — the Grassmannian a . . . c k, n c ( n 1 C dimensional vectors modulo a + c − h M n k . 1 c -planes in (tree) Amplituhedron 1 k − k 0, so that the → matrices with all positive ordered minors. The only difference with the (1) specifies a line in 1) > ) invariant notion of positivity for the matrix − b n ( matrix k ( /c /GL × n . Once again we would like to discuss the interior of this region. However in a ) → I n c n k GL × n k Note that while both Projective space is the special case of One obvious generalization of the triangle is to an ( , . . . , c ,...,Z , . . . , c 1 I 1 2 c 3 Polygons Another natural generalization ofZ a triangle is togeneral a this more is general not polygon canonically with defined — if the points positive Grassmannian is that in the columns of thegive matrices, a they positive have matrix, ac then positivity natural is cyclic preserved structure. under the Indeed, (twisted) if cyclic ( action We can also talk aboutare the just very closely related space of positive matrices “inside of a simplex” in projectivepossible space can be generalizedordering to the of Grassmannian. the The columns, only and demand that all minors in this ordering are positive: take to be a collection of into a to refer to this as “positivity” for brevity. a general projective( space, a collectionthis to ( the space of More precisely, the interior ofall a ratios triangle is associated withgeneralizations a that triplet follow, ( we willthe closure abbreviate of this the by triangle calling replaces “positivity” them with all “non-negativity”, positive. but we Including will continue JHEP10(2014)030 . n (3.4) (3.1) < t form a matrix ··· a n 2 Z × < t 1 t (3.3) ) ), with (3.2) , n m a 0 (3 > + 3 ], which is the convex hull of a M c , . . . , t 33 2 a ⊂ < a (3.6) , t 0 ) 2 a . n a > , t (3.5) with Z < a 0 a c are all vertices. Only if the 1 > I , which are positive n a = (1 a i I ,...,Z a Z m a 1 Z n m Z Z Z a form a closed polygon only if the 3 ( 1+ c , with a 1 + a m . . . c – 6 – = Z , P , 0 for ) + I I a ...Z I 2 ,..., Y > Z , n 1 a Z i a c 2 (1 3 Z c a = 1 + h = Z I + G 2 I gives us all the points on the inside of the polygon, living is a special case of positivity in the sense of the positive a I 1 ⊂ Y a Z a Z c 1 ) 1 Z a n c Z ), with h = m I Y , . . . , c 1 1 + c , ( has all positive (ordered) minors: (1 a G Z in , ranging over all I a a Z Z polygon do we have a canonical “interior” to talk about: 3). , (1 This object has a natural generalization to higher projective spaces; we can consider Now, convexity for the G points ordered points on the moment curve in This space is very closelyn related to the “cyclicFrom polytope” our point [ of view,this forcing is the just points one to way lie of on ensuring the the moment positivity curve is of overly the restrictive; n Then, the analog of the “inside of the polygon” are points of the form for fixed in We have If we jam them together to produce Having arranged for this, the interior of the polygon is given by points of theNote form that this can be thought of as an interesting pairing of two different positive spaces. matrices we have just defined.with The columns points convex don’t obviously enclose a region where all the JHEP10(2014)030 (3.7) (3.8) (3.9) . In this : I a 3 δ = 8, which P = n I a ). Z m + ), with co-ordinates ), where we can use m ). The polygon is the m k, k Z + . It is natural to restrict ( dimensional space and is ( + + m k k is even: under the twisted G k, k + (3.10) ( ) n,k,m = ( ) dimensional space. Consider G m , and the product is invariant A 1 m n m c ⊂ m, n 1 + + , . . . , k − Y + k k = 4, and we will refer to this as the positive external data in k = 1 1) ( a m I + − Z ( , . . . , k I a M Z Z for for → · = 1 ⊂ I a αa 7 n I a C Z ) determined by taking all possible “positive” C c Z – 7 – 7 m = ,Z = ) I + α Y and . . . c Y , . . . k, I 1 k, n fill out the entire ( ( + Z k, k a + ) dimensional space, 1 1 ( = 1 Z − Z G G m 1 m = 1, it is a polytope, with inequalities determined by linear c + ⊂ , α + k I k k α = 2. Another special case is = αa Y 1) ). The tree amplituhedron lives in a 4 Y C − Z , m ( 1, it is not a polytope and is more “curvy”. Just to have a picture, ( ) dimensional vectors n,k → = 1 m k > A n k + Z k is identical to the usual positive Grassmannian -planes in this ( k ), so that the external . ) transformations to set the external data to the identity matrix m,k,m m m m + + k + k A ( The case of immediate relevance to physics is We call this space the generalized tree amplituhedron We can further generalize this structure into the Grassmannian. We take positive k ( ≥ equations, while for below we sketch a 3-dimensionalturns face out of to the be 4 dimensional the amplituhedron space for GL case tree amplituhedron not trivially visualizable. For It is trivial tocyclic see symmetry, that thisfor space even is cyclically invariant if simplest case with where We then consider alinear subspace combinations of of the external data, or more explicitly external data as ( n the space of JHEP10(2014)030 ) i C j Z i, j i . In (4.1) n map is Z YZ n h . If as we Z · 0, and also i, j . . . c C > a + = c , there is always 1 m Z Y 1 are not consecutive, + c for some is stuck between i = i, j a j Z Y i n > k (4.2) YZ 0 ) in the interior of the space, h j > Z i with no positive restrictions on i i Z j +1 Z Z = 2, with i i · Z Z i C a Z , m Z a = h a Z c h Y = 1 a changes sign from being positive to negative, c a k ) is a boundary of the polygon: i – 8 – X j j a are consecutive, we get a manifestly positive sum: X Z Z = ). The reason is that for i i lies on the line ( i Z = j ’s. m i, j i Y YZ Z C + i which sum to zero! We have to take care to avoid this h +1 a i + 1, this is manifestly positive. Thus the boundaries are ’s, YZ Z Z k, k i c h ( ), but we wish to see this more algebraically, in a way that i, i 0 and G +1 i YZ 6= h Z are positive. It is however obvious that if a i ’s and the i → s Z j 0 Z Z Z i 0 for YZ > h i +1 i ) as expected. Z i ) should not be a boundary of the polygon. On the other hand, if +1 j i Z a Z Z i i Z Z Z h . But in general, this will not be projectively meaningful, and this expression won’t Of course for the polygon it is trivial to directly see that the co-dimension one bound- It is simple and instructive to see why positivity ensures that the Z Since lines ( We can see why there isordered some minors hope of the for thesome positivity of of this the sum, terms sincewill the in be this negative. sum But precisely will when be positive, but some (where aries are just thewill lines generalize ( to the amplituhedron where “seeing” is harder. So, we compute thus ( everywhere has a uniform sign, then ( one boundaries of thefor generalized the tree simplest amplituhedron. caseorder to Let of look us the at illustrate polygon thesweep boundaries the with through of idea the all already space, the letthen allowed us somewhere compute some linear combination of the happening, and in orderproperties to on avoid both “0” the on the left handprojectively side, well-defined. we obviously We need will positivity see this as a by-product of locating the co-dimension We have motivated theof structure the of “inside” the of amplituhedronneed a by for convex mimicking positivity. polygon. the We can geometricor However attempt idea there to is define aallow simpler us and to define deeper a origin region of in the 4 Why positivity? JHEP10(2014)030 ≤ = 1. ), as (4.3) k (1) on mk +1 (4.5) j + 1), we 0 Z GL ) can only j > ) = Z m i , j, j m + +1 i +1 j + Z + 1 , which lie inside i Z is respected. For k k, k j Z ( i, i Z C k, k ,Z G ( (4.4) j = 4 already for +1 0 i G ,Z m Z i > i ) is a simple and striking i Z Z is projectively well-defined. ) is on ( k +1 k a i +1 j l Z j Y · Z Z Z + 4 dimensions. j k j C ··· k ...Z Z Z Z 1 ) of the form ( j 1 = a Y +1 +1 Z i . Since dim i i Z Y is on the plane ). For the polygon, these are the cells Z Z Z Z i i ih a · k Y Z Z a Z ) is a boundary by computing i, j, k, l k, n a l h C ( -plane in a Z Z -plane ( + k c is projectively well-defined. There is no way h ) are non-vanishing. The image of these cells k k . . . c a = k G Z c 1 Z a j a . Let’s look at the case , c · – 9 – X Y j c Z a , provided the positivity of h i C X = k k , c Z i i k, m c = = , this is in general a highly redundant map. We can l

0 + = a       Y ) = 1 Y d Z a space is simply = ). To see how, note that we we can always use a a h 1 = Z Z a 0 Z → Z ; Y a ( Y 1 , . . . , a of momentum-twistor space. Accordingly, we identify a Y 1 with the usual bosonic momentum-twistor variables Z Y ( YZ n,k a α that cover the full amplituhedron, Ω 3 h a P Z n,k = T = Ω Y 4 4 a a α dα , and this geometry in turn determines the form Ω. However, once ... Z 1 1 a a = 1). Its image in α dα 5 a , α 4 a ) transformation to send + 1’st power of them vanishes. This is equivalent to saying that each entry can be k N , . . . , α Note that the definition of the amplituhedron itself crucially depends on the positivity (4 + (4) symmetry, as the usual 1 a α 7 The superamplitude We have already defined central objectsthe in associated our form story: Ω that the is treecan loosely amplituhedron, speaking together be its with “volume”. directly The extractedGL tree super-amplitude from Ω As with the polygon, the form is independent ofof the the particular external cell data decomposition. this form is in hand, it can be analytically continued to any (complex!) Now, given a collection of cells are vanishing for all but( columns and the form is For instance, consider 4 dimensional cells for written as We still have to decidenormal how bosonic to interpret variables, the wetherefore remaining have natural an to try infinite andsome number make of the extra remaining components degrees infinitesimal, of by freedom. saying that It is the top four components of the We can think ofGL the 4 dimensional space complementary to JHEP10(2014)030 ]. n,k 34 , M 19 adjacent , 17 of dimensional (7.8) ) +4) has given k a = 4 does Z pairs ) k, k (7.4) z ( ) N ( 0 (7.5) G ) ) (7.6) Y αa β ; I Z 0 C . ; (7.7) Y ( Y ) ( k Y β α N a k 4 ( | ) ρ 4 Z k δ 4 ; L ) − n,k 0 δ ; ω I Z Y k ,..., α k ; Γ ( 4 Γ 4 Y ( Y α i × n,k dα ( k, n = 1 k ( ω +4) Y k A n,k + k 4 ... ( φ Ω d ). , however, only for × G Γ 1 k N Γ 1 a k Z N δ α dα , η k 4 a ) φ ...Y . . . d space, then it is easy to show that z ) are positive co-ordinates for the cell, and ρ Z 1 1 k N ]. 4 Y Y φ -space associated with a given 4 h ’s: – 13 – 19 N Y det( ) = φ d 0 . . . d β α ... Y 1 ρ i ; Z k φ , . . . , α 1 1 Y × Y N ( k 4 α d k d ( ) = d 4 function a k δ Γ Z αa Z δ Γ , η C a Ω loop positivity in z ...Y ) = ( ) = + 4) is of the form 1 a Z 0 Y k Y → n,k h , η ; φ a k, k 4 z Y ( M ( ( are Grassmann parameters, and G k Ω = 4 n,k a . . . d δ ) are the super momentum-twistor variabes. This is precisely the formula is the associated form in η ). Then, if 1 a , and integrate over the ]. Consider the form in M k η φ k 0 4 Γ | Γ 4 4 ) is a projective Y a k, n α 0 d dα and 19 z ( , Y + ; Z 17 ... = ( ) to G Y ,...,k ( Γ 1 a Z 1 Γ 1 k ; α φ 4 Z dα δ Y . Any form on ( It is trivial to imagine what we might mean by hiding a single particle, but as we will This connection between the form and the super-amplitude also allows us to directly Now there is a unique way to extract the amplitude. We simply localize the form = Y Γ n,k The direct generalization of “convex polygons”us into the the tree Grassmannian amplituhedron. Wenatural will way? now ask This a will simple lead question: to can an we extended “hide notionsee particles” of in momentarily, positivity a the giving idea us loop of amplitudes. hiding particles is only natural if we hide have extracted from them super-amplitudesthe which connection are to manifestly the supersymmetric. Grassmannian Indeed, each shows cell much more is — manifestly the Yangian superamplitude invariant [ obtained for 8 Hiding particles where for computing on-shell diagrams (inThus, momentum-twistor while space) the as amplituhedron described geometry in and the [ associated form Ω are purely bosonic, we mulae of [ cell Γ of Ω Note that we canfurther define have weight this zero operation under for the rescaling any ( exhibit the relation between our super-amplitude expressions and the Grassmannian for- and our expression just says that Here Note that there is reallyfix no integral to perform in the second step; the delta functions fully Ω where JHEP10(2014)030 with (8.1) (8.2) C 2-planes L dimensions n ) individually, 1 ,B 1             . . . A ∗ ∗ ∗ ∗ ∗ ∗ , which is additional -plane in k . . . ,...... n ...... 3    ) and ( 2 ,A 2 (1) (2) ,B C . . . + 1 ∗ ∗ ∗ ∗ ∗ ∗ ). We can always gauge-fix the 2 D D ,A ) are taken into account, we mod . We do this simply by chopping 1 i A m 2 A B    i , ), ( ,B , . . . n 2 A 2 . . . 2 ! B ,B + 1 ,A 2 1 2 (2)    . . . C 00 0 1 0 0 0 0 1 0 0 0 , m ,A D ,B 2 1 1 ) columns into each other. i (2) (1) C A

. . . ∗ ∗ ∗ ∗ ∗ ∗ ,B at all enforce that the ordered maximal minors ,B D D , – 14 – 2 1 ,B in separate columns, the remaining minors would 2 i ! A    A . . . ,B . . . m... A ...... 2 (1) ,... C 3 D ,A . . . 1 ,A , . . . , m, A 2

2 . . . ,B 1 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , , 1 ,A  1 A 1 , 1 1 . . . A C B ,B  1 1 . . . 0 0 10 0 0 1 0 0 0 0 0 A A columns so that the matrix takes the form             2 (2) action rotating the ( ,B 2 A GL and 1 This “hidden particle” picture has thus motivated an extended notion of positivity We can now see why particles are most naturally hidden in pairs. If we had instead ,B , with all ordered minors positive. But we can also imagine a collection of 1 But the “hidden” particlespositivity leave of an the minors echo involving in allas of the well ( positivity those properties not ofof involving the this following matrix. matrices The We would now like toout “hide” the the corresponding particles columns. The remaining matrix can be grouped into the form columns we’ll label ( A particles. To pick a concrete example, suppose we have some positive matrix order to ensure that onlyout minors involving by the the pairs ( associated with the Grassmannian.C We are used to considering a hidden single particles as be positive or negativestructure depending over on and the above orderingsarbitrariness, the of we cyclic should ordering hide of particles the in external even data. numbers, In with order pairs to the avoid minimal this case. In are all positive. JHEP10(2014)030 C (9.1) . We can C ) as + 4) dimensions, ). Note that this ,C k ) C i ’s leads us to define ) dimensional space; ( k D , schematically ). First, the external D in ( Z Y ( Y k 4+ n,k,L A planes < a k ··· . living in a (4 + ) i ( < I a 1 D , are positive. (All minors must include Z ) a , schematically i ( C D . Y – 15 – ) by demanding that not only the ordered minors ): the space of 0 for L ) ) as . The data is positive ; L > Z k ( ,Y i ) k k, n i ( ( + 4; 4 + in the obvious way to include the 4+ L G a n,k,L k, k in the 4 dimensional complement of ( A are defined to live in the complement of C.Z ,..., ) ...Z G ), and we will denote the collection of ( ) i i ( 1 = ( L a = 1 L ; D Z Y I h k, n ( with any collection of the and G ) dimensional complement of C 2-planes k + 4 particles is given by the vectors L − , since the k n , . . . , n C ≥ n = 1 a in the ( , but also of Extending the map Very interestingly, it turns out that while we motivated this notion of positivity by ) C i ( We will denote the collection of ( The amplituhedron lives in together with 9 The Amplituhedron We can now givedata the for full definition ofwhere the amplituhedron hiding particles from anthat underlying can positive not matrix, be there obtained are by positive hiding configurations particles of fromthe a full positive amplituhedron. matrix in this way. We call this space extend the notion of positivity to of the matrix notion is completely permutation invariant in the D JHEP10(2014)030 ) 2 i ’s = )) ( , ) is α Y γa ( Γ (9.2) (9.3) (9.5) (9.4) Z = 1 α ; D matrix ,C Y γ ) ) ( n α Z ( ( × ) ) i (10.1) l ( ) n,k,L 0 D n,k,L Y +2 , where (which we have ; ) A k i ) ( Y i I γ ( ( ) k ) = ( L 4 α δ =2 ) consisting of all ( ) γ L C Z L in the obvious way. The ; , with the positivity prop- ) of the form i =1 -dimensions, and the C ( ) γ + 4; i n γ ( L . n,k,L L L I a Γ i k, k ) minors of the ( Y, of non-intersecting cells Γ whose Γ i Y, Z l ( ( n,k,L ) α i dα ) for the loop integrand form is G T ( ’s positive, ) +2 γa α L k -plane in n,k,L 7.4 ( + k D =1 Ω k i Y × 4( ) = , and the 2-planes are       Z l Z I ) T α i k ) ( ) ⊂ Y l φ 1 X I γ +2 i Γ i , can be taken to be non-vanishing in the first C· gives us a line ( 4 . . . ( ( 0 C k L ) Y – 16 – D i = D , ( ) = specifies a I γ a . . . d Y Z L Z       1 ; αa φ αa -plane is , such that for Y 4 C ( k C d C ) is the subspace of ) dimensional complement of = . These are the momentum-twistor representation of the Z k Z 3 ( n,k,L I α − P ). Each Ω Y k n ) = ) × i n,k,L 2 , all the ordered ( ( 0 ) in γ A L | ( ) cover the entire amplituhedron. The rational form Ω i 4 -form for the loop integrand from Ω ( L ≤ × ) ; Z L l a )2 i ( C· AB , η ≤ ) has logarithmic singularities on all the boundaries of L a . The amplituhedron is the space of all Z z = ), rational in the ; ( ). A cell decomposition is a collection ) = ( L Y L Y ; n,k ( + , being in the complement of ) i ) k i I ( ( M Γ 4( L ,...,L k, n L n,k,L ( + Ω 2-planes living in the ( = 1 G i The amplituhedron A concrete formula follows from a cell decomposition as The notion of cells, cell decomposition and canonical form can be extended to the L , . . . , α Γ 1 α where as in the previousare section the erty that for any 0 which are a positive linear combination of the external data, More explicitly in components, the and 4 entries also been calling ( loop integration variables. The analog of equation ( 10 The loop amplitudeWe form can extract the2-planes 4 Of course any cell decomposition gives the same form Ω is in images under defined by having the property that are positive. full amplituhedron.( A cell Γ is associated with a set of positive co-ordinates JHEP10(2014)030 ) ) i ) a i Z Z i ; (10.5) (10.6) (10.2) (10.3) ) ], now + 1 )( a, b, c 56 i 16 ) ( i ( γ 18 L L , 2 Y AB 0 Y AB Y, i AB j j ( Y ih ih ( ih 15 n,k + 1) n,k,L = 1 amplitude for 235) ω + 1 2 ω i k i Y j +1 ) j j 4) for each cell is i ( j , 1 Y AB 2( Z (1 i × ∩ (2 ih ) L 456) i ∩ +1 2 ) are labeled by a pair of G AB j 2( 13 Y d ) ( α ih L i , n j 2 (561) ∩ 2( + + 1) d (2 1 ) L j i + Y AB ) i i i Z 2( G matrix is just restricted to be in ih j 1( AB Y AB L (1 (123) α ) 23 (10.4) i ih D ih ihL + 1( ) AB i 1 L + 1 + 1] ih 1( Y AB Z 234) Y Y AB h L B 4 − Y ih 2 ih 2 j j i ( ) d i = ) 12 ∩ i 1( AB i i , and the 2( L ih ABd 2 ,B L – 17 – + 1; 1 ) d ih (2345) AB i ) Y AB i +1 (561) + 1 A ∩ i 1( i i 2 i ih 2( Z 1 [1 L hL ) matrices, let us consider the 1-loop 1)) i +1 L =1 ). A simple collection of these i Y AB i Y 00 ABd i

− t + + 1 + t 1 + 3 3 3 + t t ), but also many unphysical poles. The unphysical poles are associated zt w 2 4 i 3 0 0 0 xt 1 + wt 0 0 x y yw y x y 12 + ) these collection of cells do cover the amplituhedron. + y t 3 2 2 2 b 0 0 0 t 1 0 0 1 t t x wz z t w + + − 2 2 t y 2 y 1 2 t 2 Y AB ). We then choose any positive matrix t y t x t t t − − h 6 − 00 0 1 0 1 1 inside the amplituhedron. We can ask whether or not this point is contained in 1 ) t t x ,Z 0 0 0 x y 000 1 0 1 1 t − t 1 t z w Y − 0 1 00 − + w ··· 1 It is satisfying to have a definition of the loop amplituhedron that lives directly in the Note that the form shown above, associated with a BCFW term, has some physical We have checked in many other examples, for higher w − (1 + 1 w xt w w 1 , − z 0 0 − 0 0 0 0 t 1 1 0 00 1 1 ) BCFW terms can be expressed as canonical forms associated with cells of the ampli- a Z       associated with the following positive co-ordinates for the ( While it may not be immediately apparently, this is nothing but the “dlog” canonical form ( tuhedron and ( space relevant for loopthe amplitudes. loop integrand This using is recursion relations, in which contrast ultimately with traces the back to approach higher to computing poles (like with boundaries of the cellamplituhedron that themselves. are “inside” These the boundaries amplituhedron,poles, and which are not cancel spurious, boundaries in and of the the so sum are over all the BCFW corresponding terms. point one of the cells,variables by of seeing the whether cell. Doingin this just we one find of that thesecells). every cells point (except The in of cells the course taken for amplituhedron together points is therefore on contained the give common a boundaries cellulation of of different the amplituhedron. positive, and all the orderedall 3 the positive variables, whichspace. under Remarkably, we the find that theseThis cells can are non-overlapping, be and checked cover( the directly entire in space. a simple way. We begin by fixing positive external data One can easily check that for all variables positive, the bottom row of these matrices is This exercise can be repeated withare all 16 BCFW terms. The corresponding ( JHEP10(2014)030 ] 0. 19 is a 1 i → Y ) j ( matrix is matrix to C AB C ) i 0; in the loop- ( AB i → h +1 0, exactly as needed j Z j 0, and Z i → +1 +1 i j i → Z Z i j Z in four dimensions. The BCFW +1 Z h 2 +1 ) ). We can e.g. assume that i Z +1 i j AB i i AB and the amplitude, it is obvious that the Z h ( Z k , j 1 Z ) n,k,L +1 ...Y i AB 1 Z Y – 19 – i h ), and thus that the top row of the Z +1 j = 8 tree amplitudes, and arrives at the form we are ,Z j , n ,Z = 2 +1 i k ,Z i Z 0. ) is on the plane ( k i → ...Y + 1 1 space, and we can find a cell decomposition for it directly, yielding the form Y 2 ) AB i i AB k ( Y , plane ( 1 ) The factorization properties of the amplitude also follow directly as a consequence of Given the connection between the form Ω We have understood how to directly “cellulate” the amplituhedron in a number of k ··· tree amplitudes. Consider the simple case of the 2-loop 4-particle amplitude. We are 1 Y AB positivity. For instance,the let us considerlinear the combination boundary of of ( only the non-zero tree in amplituhedron these where columns. But then, positivity remarkably forces the first (co-dimension one) poles“faces” of of the the amplitude amplituhedron. areforces For trees, associated these we with faces have the already tofor co-dimension seen be locality. that, one precisely The remarkably, positivity analog where h statement for the full loop amplituhedron also obviously includes level integrand, we can also haveUnitarity poles is of reflected the in form what happens as poles are approached, schematically we have [ Locality and unitarity are encoded intiful the way. positive geometry As of is thestructure amplituhedron well-known, in of locality a the beau- and integrand unitarityonly for are allowed scattering directly singularities at amplitudes. reflected tree-level in should In the occur momentum-twistor when singularity language, the however simpler, without an obvious connection to the BCFW expansion. 11 Locality and unitarity from positivity other examples, and stronglyto suspect do that this. there willparticularly nice, be The canonical a BCFW cellulations general decomposition ofgives understanding the of a for tree tree cell how amplituhedron. amplitudes decomposition. Loop seems level BCFW The to also “direct” be cellulations associated we with have found in many cases are after a form inapproach the begins space with of the interested two in 2-planes after ( taking two( “forward limits”. But the amplituhedronwithout lives having directly to in refer the to any tree amplitudes. n JHEP10(2014)030 matrix in C matrix n × ! R ↑ ↓ 1 k 0 + 1 − ] of the NMHV tree ampli-           35 0 0 0 0 n n x ...... ! − (11.1) n ) n x 0 0 0 0 R , which adds three degrees of freedom n − A ...... n. . . n y + 1 1. This factorized form of the . . . y 2 ∗ ∗ and y − 2 j j ...... y . . . n y k n 1 0 0 0 0 1 – 20 – 2 y 1 2 0 0 = y y B R ], the forward limit in momentum-twistor space is y L ...... k 1 A 1 0 1 2 A particle one-loop MHV amplitude. On the boundary 18 y y + 0 0 0 0 ( αy A n

L k + + 1 αx matrix to the forward limit [ B + 1 between y i i ∗ ∗ n D matrix has the form           A A D AB L y x ↓ ↑ k such that

R 0, the , k L k i → 1 AB n h , α A The connection of this We can similarly understand the single-cut of the loop integrand. Consider for con- , x n and first we “add”x particle we start with the tree NMHV amplitude, associated with the positive 1 tude is simple.represented In as the language of [ turn implies that on thisamplituhedra boundary, in the exactly amplituhedron the does way “split” required into for lower-dimensional the factorizationcreteness of the the simplest amplitude. casewhere of the for all possible “factorize” in the form JHEP10(2014)030 and and ) which (12.1) (12.2) i ( +1 B n D c , y − A 1 . An example c as j b (12.3) we are then left ~ ) , α, y i 0 ( → A → x < D 1 i c 2 minors of ) b ~ A, B j × w , the line directed from − ! 0. Geometrically this just i i, j n w n < x )( j ) − j y b ~ − − i ! ...... n . . . y i y i b ~ ( i w 2 z · ! y − ) + ( ) = 3 is j j ) ) 0 1 z i j ~a L ( 1 ( i i columns. Chopping off 1 2 1 0 y − y x − D D i i 1 0 z – 21 – ~a A

)(

A, B j αy A loops. We can parametrize each x = ). The points are in the upper quadrant of the plane. det i ) + L − αx i ( i , w B i x y D ( z 1, which means shifting column 1 as , A A , = ( AB y 0 x i b + 1 ~

, > ) n i i columns have precisely four degrees of freedom (2) acting on the , y , w matrix on the single cut. This shows that the single cut of the loop i must be arranged so that for every pair i + 1). This gives us the matrix x n b D , z GL ~ i A, B = ( ~a, , y i i ~a x is pointed in the opposite direction as the one directed from 4 minors j × ~a → i of an allowed configuration of such points for We can rephrase this problemsional in vectors a simple, purelyThe geometrical mutual way by positivity defining conditionmeans two is dimen- that just the ( are positive. This translates to In this simple case thethe positivity 4 constraints are just that all the 2 12 Four particles atLet all us loops briefly describelevel, the for simplest four-particle example scattering illustrating at the novelties of positivity at loop note that the the we can remove by precisely with the integrand is the forward limit of the tree amplitude, exactly as required by unitarity. removing column ( and we finally “merge” ~a JHEP10(2014)030 0 for (12.5) ’th line → L i w . To pick a . ,...,L ): i + perm dy + 1 i 2 1 z just have to be positive. dz L i ), so that the i 1 2 − y = dx Z ). The positivity conditions 1 i 1 I 2 z Z Z Q ’s we have are ordered in some 1 ... x Z . The L 0 for L z (12.4) 0 → − > z 1 I 1 < y − ) ··· L ], it is easy to find a cell decomposition j z z > 26 L 1 − 1 z i z 2 lines to intersect (12) and 2 lines to intersect 1 z − – 22 – − )( L ), so that dimensional space directly gives us the integrand j L 4 x x 1 Z L − 3 − -loops. i Z L x x L ( ... 1 x 1 ]. Here we will content ourselves by presenting some especially − 2 26 x . Then we must have 1 L 1 x L 1 < x y ··· ... and others intersects ( 1 < 1 1 y 1 x 0. This corresponds to having all the lines intersect ( ,...,L We can continue along these lines to explore faces of the amplituhedron which deter- Let us start by considering an extremely simple boundary of the space, where all the As we will describe at greater length in [ Now, we know that the final form can be expressed as a sum over local, planar diagrams. → i = 1 mine cuts to allway. For loop instance, orders suppose that thati some are of difficult the (if linesconcrete intersect not interesting ( example, impossible) let to choose (34). derive We in can any further other specialize the geometry and take more cuts by making the contribute to this cut:have the above “ladder” from diagram. positivity. The corresponding cut is precisely what we Now, this cut is“local” particularly expansions simple of to the understand integrand from — the there point is of only view only of local the diagram familiar that can possibly which is trivial to triangulate.way, say Whatever configuration of The associated form is then trivially (we omit the measure cuts systematically in [ simple but not completely trivial examples. w then simply become for the full spaceapproach “manually” to at understanding low-loop the orders.As geometry an that interesting We warmup might suspect to crack the thereof the full is the problem, problem a we four-particle at can more amplituhedron. all investigatecertain systematic lower-dimensional loop “faces” cuts Cellulations order. of of the these integrand, faces at corresponds all to loop computing orders. We will discuss many of these faces and for the four-particle amplitude at This makes it allproblem do the we more reference to remarkableis diagrams that one of of any no-where sort, many in planar that or the emerges not! from definition Nonetheless, positivity. of this property our geometry Finding a cell decomposition of this 4 JHEP10(2014)030 : σ , we → ∞ L (12.8) (12.9) (12.6) is just 1      i space is ’s where − 5 y L x L , there is s /w 0 1 ) , w x, y x i , the second < σ 5 1 − 1 i L x − − L < σ L − z 1 < σ < σ 3 L i x σ 1 ( < x σ − 3 ) L z 1 i 1 σ ) − x 0. Let us also take the − l (12.7) 2 − L ) σ 1 z 3 L x − 1 y → ) − x 1 − < x 1 2 − ) 1 1 L y − − 1 1 z L z L ( 4 i − − l ) space is just σ x L /w ( x i x i ... /W /W ( z ) ; ) ) ) i x, y 2 1 3 1 L x L > z =1 − x = 5, and an ordering for the − l Y − i x , and the only constraint on L i L i − y 1 L 1 − y . Note that if z y − i 1 − L 4 i , then both inequalities are satisfied and > z − − L x , y x L L i ( 1 ’s, we get the final form. For general 4 ( = 0. W , w x x x 1 − ··· ) 1 ( L − , the first inequality is trivially satisfied and i − L − 0 1 w z ’s, and the corresponding form in < x L i x – 23 – σ − > y y y i )( x L i ( 1 ) − 2 y 5 ( 1 − <σ = , z > x x 1 2 − L so they are in increasing order; then the positivity 1 < x      i − . Thus, given any ordering for all the z L − − − L 2 x L L x I 1 σ L 3 − ( ; − =1 x z − 2 i Y x L 3 ; /w L − i > ) < x − )( L x x X × i 2 L L y z x x <σ 1 ( ··· 2 ··· − − fixed. − − L < < . If 5 i 1 1 L 1 . The corresponding form in ( × 1 x σ x W 3 z − ; ’s is associated with the form − ( x L σ )( i z L 1 4 0. Finally if ) W x < x /W 5 > z > +1 ) is trivially satisfied for positive ≡ − l i z i i 2 1 i x 1 y y y x − < x − L − )( 2 l 1 z 1 /z ( x 1 − 2 < x L − − − L x =1 4 l Y 4 ( L w 1)’st line to pass through the point 4 — this corresponds to sending x This gives us non-trivial all-loop order information about the four-particle integrand. We can again label the ( > < x 1 − x 1 i L where we define for convenience The expression has aand feature loop familiar level from amplitudes. BCFWsum. recursion This Each relation result term can expressions has be for checked certain tree against “spurious” the cuts poles, of which the cancel corresponding amplitudes in that the are By summing over allcan the simply possible express orderings the result (again omitting the measure) as a sum over permutations an associated set oftrivially obtained. inequalities For on instance, the considerx the case The interesting partinequality of on the spacey involves we just have we just have This space is alsoThe trivial ordering for to the triangulate, but the corresponding form is more interesting. and ( with conditions become pass through the point 3 — this corresponds to sending JHEP10(2014)030 , – 0 ,L 36 0 n,k,m ,k 0 A n . How- L A and loop order n, k . It is obvious that the amplituhedron is contained in the “faces” of m n,k,L A separately for every – 24 – n,k,L A , we can also generalize the notion of “hiding particles” m . The objects needed to compute scattering amplitudes for any > L 0 ) and can be defined for any even m > k, L + 0 . . m k, k ( = 4, of relevance to physics, is contained amongst the faces of the object defined G → ∞ > n, k m 0 If we consider general even In this vein it may also be worth considering natural mathematical generalizations of n ]. The inspection of the available local expansions on this cut does not indicate an lives in with for higher in an obvious way: adjacent particles can be hidden in even numbers. This leads us ever, a trivial feature offor the geometry is that number of particles to alln, loop k, orders L are thus contained in a “master amplituhedron” with the amplituhedron. We have already seen that the generalized tree amplituhedron pression obtained in this way perfectly matches the amplituhedron computation of13 the cut. Master Amplituhedron We have defined the amplituhedron with intricate numerator factors. Ifall all physical terms propagators, are the numerator combined has with 347 a terms. common denominator Needless of to say, the complicated ex- 40 obvious all-loop generalization, norbe does expressed it in betray the one-line anyof form the hint given cut that above. is that given For the instance as final just a result at sum 5 can over loops, diagrams, the local form but now there are non-trivial numerator factorsof that propagators. don’t trivially The follow full from the integrand structure is available through to seven loops in the literature [ available in “local form”. The diagrams that contribute are of the type JHEP10(2014)030 in = 4 = 4 Y m m 2)-planes) ] have to do − -planes ). The k 15 m , Z = 4 dimensional ( (( 2 14 2 − ), of m 2 − m − m m ,...,L 2 ...L m,L ; planes) in ,...,L 4 − n,k A (2 ,L 2 L L ; (4-planes), m 4 L + k, k ( G – 25 – together with Y , schematically: (2-planes), Y 2 L ). We can call this space planes) 2 − , with the obvious extension of the loop positivity conditions − m k Z C· ) of ( = L ,...,L 4 Y ] gives us a wonderful description of the boundary correlators of this ,L +4; , we can consider a space 2 41 L Y ; k, k ( G k, n ( dimensional complement of G ) dimensions, together with m to m C + k gravity duality [ kind, and givesduality a is first still working anphysical example language, equivalence of with between emergent time ordinary runningduality space inapplicable from physical and to infinite systems our gravity. past described universe to for infinite However, in cosmological future. this standard questions. This Heading makes back the to the early cases like asymptoticallyobservables, AdS that or can be flat measured byof spaces, infinitely space-time: large we boundary apparatuses correlators still pushed for tothat have AdS the space no precise boundaries and precise the quantum S-matrix observablessuggests mechanical for can flat that space. be there The associated fact shouldany with reference be the to a inside the way of of the interior space-time computing space-time strongly these at boundary all. observables For without asymptotically AdS spaces, gauge- mechanics and gravity togethermechanics make forces it us impossible toand to divide a the have finite world local system in observables.of being two space-time, observed. pieces Quantum gravity — However for makes anit any it infinite also observations measuring impossible becomes made apparatus to in heavier, a make and finite the collapses region apparatus the observation arbitarily region large, into since a black hole. In some 14 Outlook This paper has concerned itselfHowever the with deeper perturbative motivations scattering for amplitudes studyingwith in this some physics, gauge fundamental articulated theories. challenges in of [ quantum gravity. We have long known that quantum on amplituhedron is again justsee whether a this particular larger facegeometry space relevant of has to any this physics. interesting object. role to play It in would understanding be the interesting to Once again we have considering complement of ( in the to a bigger space in which to embed the generalized loop amplituhedron. Instead of just JHEP10(2014)030 ]. This perhaps indicates the 42 , whose “volume” gives us the scat- n,kL ]. We have given a formulation for planar A 19 – 17 – 26 – ] and pursued in [ 15 , 14 In this paper, we have taken a baby first step in this direction, along the lines of the pro- We may be in a similar situation today. If there is a more fundamental description of Understanding emergent space-time or possible cosmological extensions of quantum = 4 SYM scattering amplitudes with no reference to space-time or Hilbert space, no to accommodate both, in the new picture they emerge together from positive geometry. ably generalized to the Grassmannian,“hiding” and particles, with leads an us extended to notiontering the of amplitudes amplituhedron positivity for reflecting a non-trivialis interacting also quantum fascinating field that theory whileare in in in the four tension usual dimensions. with formulation It each of other, field theory, necessitating locality the and introduction unitarity of the familiar redundancies gram put forward in [ N Hamiltonians, Lagrangians or gauge redundancies, nono path mention of integrals “cuts”, or Feynman “factorization diagrams, channels”,sented or a recursion new relations. geometric We have question,remarkable instead that to pre- such which a the simple picture, scattering merely amplitudes moving are from the “triangles” to answer. “polygons”, It suit- is description is unlikely to directlyrather reproduce match the on usual formulation to ofderived some notions. field new theory, Finding but formulation such must reformulations of ofus the standard for physics physics the might where then transition better locality to prepare and the unitarity deeper are underlying theory. as a natural deformation, via the path integral. physics where space-time anddon’t perhaps appear, even then the usual evenneglected formulation in and the of the physics quantum limit reduces mechanics where to perfectly non-perturbative local gravitational and effects unitary can quantum field be theory, this world is quantum-mechanical and notof deterministic, quantum and mechanics for can’t this immediately reason,formulation land of the on classical classical Newton’s physics limit where laws, determinismleast but is action must principle not match formulation a to is central some thus butlaws, much derived and closer notion. gives to a quantum The better mechanics than jumping Newton’s off point for making the transition to quantum mechanics hidden in the structuredeterministic, of there classical is a mechanics completely itself.of different the formulation While of principle Newton’s classical of laws mechanicsthese least — are very in action manifestly different terms — starting which pointsto is leading classical not to manifestly physicists, the deterministic. but same physics today The was existence we somewhat of know mysterious why the least action formulation exists: the mechanics will obviously berectly attacking a the tall quantum-gravitationalconfronted. order. questions But where The there the isradical most relevant another step obvious issues strategy taken avenue that must in forcal takes be determinism the progress some was transition inspiration is lost. from from di- There the classical is similarly to a quantum powerful mechanics, clue to where the classi- coming of quantum mechanics bang singularity. More disturbingly,of even our at universe, late we times, onlydivision have due of access to the to the a world accelerated finite intoto number “infinite” expansion talk of and about degrees “finite” precise of systems,need observables, freedom, required seems for and by an to thus quantum extension the be mechanics of impossible quantum [ mechanics to deal with subtle cosmological questions. universe, an understanding of emergent time is likely necessary to make sense of the big- JHEP10(2014)030 ] ) L 51 ; ) will L k, n ; ), which ( + , n G ]. k, n ( +2 48 + k – ( G 45 G ]). Optimistically, ] is most naturally ]. In this way, the 50 19 matrix positive seems , 19 . It would also clearly L n ; 3)’s, building up larger 49 , × n,k k (2 A G ] (see also [ 19 3)’s and minors of a ], along the lines of [ , k (1 44 × , G k 4 ] gives hope that the necessary understanding 19 – 27 – )[ ]. Remarkably, this extremely natural mathematical , n ]) “little” 19 43 + 2 k ( is given directly by construction as a sum of “dlog” pieces. G . This form was motivated by the idea of the area of a (dual) ], we will give strong circumstantial evidence that such such an n,k,L ) that cellulate the amplituhedron, and those of 27 n,k,L k, n ( G . However in [ n,k,L We also hope that with a complete geometric picture for the integrand of the amplitude Note that the form Ω While cell decompositions of the amplituhedron are geometrically interesting in their A great deal remains to be done both to establish and more fully understand our our story suggests thatan this important positive way. structure The must strikingis be appearance an reflected of example in “cluster of variables” the this. for final external amplitude data in inin [ hand, we are now positioned to make direct contact with the explosion of progress in to which the full amplitude (rather than just the fixed-orderThis loop is a integrand) highly is non-trivial the propertythe answer. of on-shell the diagram integrand, representation made of manifest thethe (albeit amplitude great less [ simplicity directly) of in this formand should arriving allow at a new the picture final for amplitudes. carrying out The the crucial integrations role that positive external data played in a completely invariant definition for Ωhave free an of analog the of need for thefor any notion Ω triangulation. of We “dual do amplituhedron”, notexpression and yet should also exist. no integral Onthe representation loop a integrand related at note, any while fixed we loop order, have we a still simple don’t geometric have picture a non-perturbative for question these ideas and Witten’s twistor-string theory [ own right, from thedetermining point the of form Ω viewpolygon. of For physics, polygons, we we have another need definition of them “area”, as only an as integral, and a this gives stepping-stone us to as “non-reduced” diagrams in can be reached. Havingvery likely control be of necessary the tobe properly cells very understand and illuminating the positive to cellulation co-ordinates finddata an for in analog the of original the twistor amplituhedron, variables.This built might around also positive shed external light on the connection between natural operation of decomposinga the vivid amplituhedron on-shell scattering into picture pieces inan the is original analogous ultimately space-time. understanding turned Moving of into to all loops,— possible we we don’t cells don’t have of yet know theboundaries how extended to and positive systematically so space find on, positive though co-ordinates, how certainly to the think on-shell about representation of the loop integrand operation translates directlygluing to together the elementary physical three-particleformulated amplitudes. picture in of This story the building ofpicture original on-shell [ is twistor formulated diagrams in space momentum-twistor from space.between or the At cells tree-level, momentum of there is space, agive direct while connection the the corresponding amplituhedron on-shell diagram interpretation of the cell [ conjecture. The usualunderstanding positive all Grassmannian possible ways has to make a ordered daunting very at rich first, but cell the structure.gluing key is together The to (“amalgamating” task realize [ that of positive the matrices “big” from Grassmannian smaller can ones be [ obtained by JHEP10(2014)030 ]. ]. 58 = 4 69 – , N 56 68 ], so it is = 4 SYM, 61 N ]. A particularly 55 – 52 ], the connection between on-shell 19 – 28 – ]). As already mentioned, one of the crucial clues 67 – ]. The amplituhedron has now given us a new descrip- 62 make all the symmetries manifest. This is not a “du- = 4 SYM, such identities, of a fundamentally non-planar 70 infrared divergence of the (log of the) amplitude gives the N 2 does = 4 SYM and the positive Grassmannian have a close analog N dualities, that are impossible to make manifest simultaneously in identities with this character, such as the BCJ relations [ ] can be realized at the level of the amplituhedron. -duality [ U T 60 , more 59 = 4 SYM amplitudes which does not have a usual space-time/quantum = 4 SYM amplitudes are Yangian invariant, a fact that is invisible in the N N Planar On-shell diagrams in superstring scattering amplitudes? mechanical description, but ality” in thetheories usual with sense, familiar since physicalnew interpretations. we mathematical are We structures are not forphysical seeing identifying representing ideas, something an the rather but equivalence different: with physics between all without symmetries reference existing manifest. to standard Might there be an analogous story for conventional field-theoretic description in terms ofin amplitudes the dual in space. one space We have orhas become Wilson a accustomed loops rich to spectrum such striking of factsconventional in string string perturbation theory, which theory.SYM Indeed is the just fermionic Yangian symmetrytion of of planar planar of the external data. Itsatisfy is only hard have to believe a thateven geometric these richer “triangulation” on-shell structure objects interpretation beyond and the inprovides the planar the a identities limit planar they strong have case, no impetus geometric while to interpretation search the at for all. a This geometry underlying more general theories. BCFW terms. We have finallysimple come reflections to of understand amplituhedron thesewe representations geometry. encounter and even As identities as weIndeed move even sticking beyond to planar planar origin, give rise to remarkable relations between amplitudes with different cyclic orderings of scattering amplitudes are alsoand widely at available — the at very loop leastprogress level for from for planar other gravitational gauge points tree theories, of amplitudesleading view (where to [ the there amplituhedron has was been theplitudes, much myriad with of recent equivalences different BCFW guaranteed representation by of remarkable tree rational am- function identities relating diagrams and the Grassmannian isthe valid building-up for any of theorythree-particle in more amplitudes. four complicated The dimensions, connection reflecting on-shelluniversal with only processes the for positive any from Grassmannian planar in gluingon-shell theory: particular form together is only differs the the from measure basic theory on to the theory. Grassmannian Furthermore, determining on-shell BCFW the representations diagrams given in [ with on-shell diagrams innatural ABJM to theory expect and the anany positive analog of null the of Grassmannian ideas the [ metry, in amplituhedron and this beyond for paper the ABJM to planar as limit? extend well. to As other explained Should in field [ we theories, expect with less or no supersym- promising place to startloop forging orders. this connection As we iswhile noted, with the the the coefficient positive four-particle of geometry amplitude thecusp problem log anomalous at in dimension, all this famously case determinedAnother is using natural especially integrability techniques question simple, in is [ how the introduction of the spectral parameter in on-shell using ideas from integrability to determine the amplitude directly [ JHEP10(2014)030 ] , , 06 (1986) 56 (2011) 058 ]. JHEP , ]. 07 SPIRE hep-ph/9403226 IN Commun. Math. SPIRE [ , ][ JHEP IN , ]. ][ point gauge theory Phys. 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Attribution License ( Thomas, Pedro Vieira, Anastasia Volovich,able Lauren Williams discussions. and Edward Our Wittendebt research for of in valu- gratitude this to Edward areaJake Witten, Bourjaily, over Andrew without the Hodges, whom past andported this many especially by work Freddy years the Cachazo would owes and Department not an of have enormous Energy been under possible. grant N. number A.-H. DE-FG02-91ER40654. is J.T. sup- is We thank Zvi Bern,ung, Jake Pierre Bourjaily, Deligne, Freddyhannes Cachazo, Lance Henn, Simon Dixon, Andrew Caron-Huot, Hodges, JamesKosower, Yu-tin Clifford Drummond, Huang, Bob Che- Jared Sasha MacPherson, Kaplan, Juan Goncharov, GregoryAlex Maldacena, Korchemsky, Song Postnikov, David Lionel He, Amit Mason, Sever, Jo- David Dave McGady, Jan Skinner, Plefka, Mark Spradlin, Matthias Staudacher, Hugh Acknowledgments JHEP10(2014)030 , 04 ]. , A 44 JHEP ]. , (2013) ]. SPIRE JHEP , IN [ 05 ]. , SPIRE , (1980) 333 , to appear. J. 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