JHEP10(2014)030 Springer October 6, 2014 : September 4, 2014 September 8, 2014 : : 10.1007/JHEP10(2014)030 Published Received Accepted doi: erent starting point. In this note we ff 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, E-mail: 91125, 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 geometry. and — 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- amplitudes role existence but where are locality derived and consequences unitarity from do a not di play Nima Arkani-Hamed The Amplituhedron JHEP10(2014)030 8 1 5 9 4 16 13 15 11 21 25 19 12 24 ) L ; ) k, n ( Z ( + ]. 7 G – n,k 1 A –1– ) Z ( n,k,L A loop positivity in = 4 SYM has been used as a toy model for real physics in many → N positive Grassmannian (tree) Amplituhedron → → 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, limit. astonishingly 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 complexity possible, 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 , erent 24 ff 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 23 ], associated with a dual 18 9 , – ] amplitudes, and represents , 19 why 8 16 20 imposing , 18 = 4 SYM as in the real world. N ]. Feynman diagrams conceal this marvelous ] and loop [ erent formulation of the physics, where locality –2– 7 13 ff , 6 ]) are remarkably connected with “cells” of a beautiful 22 = 4 SYM was pursued in [ ] pointing towards a deeper understanding is that the on- , 24 N 21 , , to determine the full integrand. This is unsatisfying, since we 19 16 , 17 used erent ways as well. The on-shell diagrams satisfy remarkable identities ]. ff erent ways, and so the final amplitude can be expressed as a sum of on- 19 ff = 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 di Planar eomorphisms [ ff solved in many di shell processes in di — now most easily understoodnian — from that their can association be used witha to cells remarkable establish observation these of 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 to amplitude, 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 di 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. infinite by 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 processes. three-particle which 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 context along as these ofical lines a derived was and planar features initiated mathematical in from [ understanding a of di scattering amplitudes [ interpretation of scattering amplitudessuperconformal symmetry as combines a with the supersymmetricinfinite ordinary Wilson dimension conformal symmetry “Yangian” loop to symmetry [ generatestructure [ precisely an as a consequence ofthe making Yangian locality symmetry and is unitarity manifest. obscured in Foras instance, either a“scattering one amplitude” of 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 , 27 a , λ are (four) grassmann 26 a η . Here .The˜ ˙ A a ˜ λ erent “triangulations” of the space ,...,n A a ff λ =1 = a ˙ A A a particle scattering amplitudes (for an excel- for p ⟩ n a ˜ –3– η , a ˜ λ , a λ | ]. 28 = 4 SYM. The discussion will be otherwise telegraphic and N erent natural decompositions of the amplitude into building blocks. ff ]) 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 Inmuch of 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 amplituhedron 2-form scattering 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 of the positivity positive toloop Grassmannian include external by order kinematical extending further data. generalizesidea The the of full 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 then a second extend generalization simplex this is into in the to Grassmannian. projective move This from space, gives us which the corresponding to di object 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 of this more a the poly- 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” di in “some space”, oped in [ JHEP10(2014)030 ) i ( γ L (1.3) (2.2) (1.1) (2.1) (1.2) .The where # I 3 a a λ µ 4) symme- ,Z | I 2 " ,Z = I 1 a ) Z z . a , 4 η | 1 , 3 1 a − P − z a ”, and we will use this a ( µ η ⟩ ], with ⟩ n,k AB 24 ] +1 +1 M 32 as “ × – aa aa 2 ⟨ ) ⟨ 30 lines in momentum-twistor space L a I 3 on it, which we can collect as , + ˜ η ⟩ denotes totally antisymmetric con- + ⟩ 1 Z a ) L a i a ⟩ 3 L λ µ c η is invariant under the SL(4 2( +1 ⟩ a ⟩ +1 as ⟩ ) 1 ... L 1 + ) 1 ⟨ , a i ! n,k I ( ) n form is [ 2 aa − ( aa ˜ − i λ ⟨ 8 L , Z ⟩⟨ a 1( ⟩⟨ a 0 M L δ ; a 2 a ) a a c L λ ... a 1 > η ) can be taken to live in 1 ⟩ ˜ λ , +1 –4– + a +1 a a a − η − c I a 12 1 a z λ , ⟨ ⟨ a ⟨ ( a Z a ⟨ a . We can write ⟨ .The4 1 z + + c M ! n,k ,L ( = +1 4 +1 a M a δ I is cyclically invariant. It is also invariant under the little ··· µ η ), so ( ⟩ ⟩ , Y a a a η n,k )= ’s is , 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 ⟨ ⟨ ˜ 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 η ϵ , n,k can also be thought of as a 2 plane in 4 dimensions. In previous work, → , + 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 η 3. The interior of the triangle is a collection of points of the form L , , 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) (2.3) c -tuple n vertices ), which ,...,c (1), with 1 n c k, n ( /GL + ) ). 3 k erence with the 0. The M ( ,c ff ) — 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 (1), with a ) ) dimensions, or a point in 1 k, n ⟩ 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 − ⟨ M nk . 1 c -planes in (tree) Amplituhedron 1 k − k 0, so that the → matrices with all positive ordered minors. The only di (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,c a 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 columns, The 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, ( wethe will 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 (3.6) (3.2) (3.4) (3.1) (3.5) (3.3) + 3 ], which is the convex hull of a M c ,...,t 33 2 a ⊂ ,t with Z I , which are positive n a =(1 a ⟩ 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 ⟩ a c 2 (1 3 Z c a =1 + ⟨ = 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 ⟨ = 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 di 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) (3.10) .Inthis : 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 + ( ) 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 a → · =1 ⊂ I a α 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 α m = 1, it is a polytope, with inequalities determined by linear c + ⊂ , + k I k k α a = 2. Another special case is = α 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 the dimensional 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 ⟩ ) C j Z i, j i .In (4.1) (4.2) n map is Z YZ n ⟨ . 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, j a j Z Y i n>k YZ 0 ) in the interior of the space, ⟨ j > Z i with no positive restrictions on ⟩ ⟩ Z j +1 Z Z = 2, with i i · Z Z i C a Z ,m Z a = ⟨ a Z c ⟨ Y =1 a changes sign from being positive to negative, c a k ) is a boundary of the polygon: ⟩ –8– * j j a are consecutive, we get a manifestly positive sum: * Z Z = ). The reason is that for i i lies on the line ( ⟩ Z = j ’s. m i, j ⟩ Y YZ Z C + i which sum to zero! We have to take care to avoid this ⟨ +1 a i + 1, this is manifestly positive. Thus the boundaries are ’s, YZ Z Z k, k i c ⟨ ( ), but we wish to see this more algebraically, in a way that i, i 0 and G +1 i YZ ̸= ⟨ Z are positive. It is however obvious that if a i ’s and the ⟩→ s Z j ′ Z Z Z i 0 for YZ > ⟨ ⟩ +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 ⟨ . 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 of hope 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 all the already space, the letthen allowed us somewhere compute some linear combination ofhappening, the 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 there idea to is define aallow simpler us and to define deeper a origin region of in the 4 Why positivity? JHEP10(2014)030 ≤ = 1. ), as (4.5) (4.3) (4.4) k (1) on mk +1 j + 1), we 0 Z GL ) can only j > )= Z m ⟩ ,j,j m + +1 i +1 j + Z +1 ,whichlieinside i Z is respected. For k k, k j Z ( i, i Z C k, k ,Z G ( j = 4 already for +1 0 i G ,Z m Z i > i ) is a simple and striking ⟩ Z Z is projectively well-defined. ) is on ( k +1 k a ⟩ +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 .Sincedim i i Z Y is on the plane ). For the polygon, these are the cells Z Z Z Z i i ⟩⟨ a · k Y Z Z a Z ) is a boundary by computing i, j, k, l k, n a l ⟨ C ( -plane in a Z Z -plane ( + k c is projectively well-defined. There is no way ⟨ ) 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– * Y j c Z a , provided the positivity of ⟨ i C * = k k ,c Z i ⟩ k, m c = = , this is in general a highly redundant map. We can l Z k ( ,Y ⟩ ) 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 ⟨ k, n ( with any collection of the and G ) dimensional complement of C 2-planes k + 4 particles is given by the vectors L − ,sincethe 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 definitionwhere of 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 = )) ( , a )is α Y γ ( Γ (9.2) (9.3) (9.5) (9.4) Z =1 (10.1) α ; D whose matrix ,C Y γ ) ) Γ ( n α Z ( ( × ) ) i l ( ) n,k,L 0 D Ω 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 Ω Γ i k, k ) minors of the ( Y, of non-intersecting cells Γ i Y, Z l ( ( α n,k,L ) α i d ) for the loop integrand form is G T ( Ω ’s positive, ) a +2 γ α L k -plane in n,k,L 7.4 ( + k D =1 Ω k i + × 4( ) = , and the 2-planes are ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , l Z I ) T α i k ) ( ) ⊂ Y l φ 1 * 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 a L Z ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 ; α a φ α -plane is , such that for Y 4 C ( k C d C ) is the subspace of ) dimensional complement of is associated with a set of positive co-ordinates = . These are the momentum-twistor representation of the , k Z 3 ( n,k,L I α − P Γ ). 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 i 4 -form for the loop integrand from ( L ≤ × ) ; Z L l a )2 i η ( C· AB , ≤ ) 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 α 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 ,..., 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 is in images under defined by having the property that are positive. full amplituhedron.( A cell JHEP10(2014)030 ) ) ⟩ ) a ⟩ Z Z ⟩ ; (10.5) (10.6) (10.2) (10.3) (10.4) ) ], now +1 )( a, b, c 56 i 16 ) ( i ( γ 18 L L , 2 Y AB 0 Y AB Y, ⟩ AB j j ( Y ⟩⟨ ⟩⟨ ( ⟩⟨ 15 n,k + 1) n,k,L = 1 amplitude for 235) ω +1 2 ω ⟩ k ⟩ Y j +1 ) jj 4) for each cell is i ( j , 1 Y AB 2( Z (1 ⟩× ∩ (2 ⟩⟨ ) L 456) i ∩ +1 2 ) are labeled by a pair of G AB j 2( 13 Y d ) ( α ⟩⟨ L i ,n j 2 (561) ∩ 2( + + 1) d (2 1 ) L j i + Y AB ) ii i Z 2( G matrix is just restricted to be in ⟩⟨ j 1( AB Y AB L (1 (123) α ) L 23 i ⟩⟨ D ⟩⟨ ⟩⟨ + 1( ) AB i 1 L +1 + 1] ⟩⟨ 1( Y AB Z 234) Y Y AB ⟨ L B 4 − Y ⟩⟨ 2 ⟩⟨ 2 jj ⟩ ( ) d i = ) 12 ∩ i 1( AB i i , and the 2( L ⟩⟨ ABd 2 ,B L – 17 – + 1; 1 ) d ⟩⟨ (2345) AB i ) Y AB i +1 (561) +1 A ∩ i 1( ii 2 i ⟩⟨ 2( Z L 1 [1 L ⟨ ) matrices, let us consider the 1-loop 1)) i +1 L =1 ). A simple collection of these i Y AB i + ′′ ABd i ) 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 a of an allowed configuration of such points for We can rephrase this problemsional in vectors a simple, purelyThe geometrical mutual way positivity by 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” ⃗ JHEP10(2014)030 0 for (12.5) (12.4) ’th line → L i w .Topicka . ,...,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 / ’s we have are ordered in some 1 ... x Z .The L 0 for L z 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 cult (if not impossible) to derive in any other − – 22 – − )( L ffi ), 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 .Thenwemusthave 1 L 1 x L 1 z =1 − x = 5, and an ordering for the − l + − 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 y y y i )( x L i ( 1 ) − 2 σ y 5 ( 1 − < = ,z >x x 1 2 − L so they are in increasing order; then the positivity 1 ) z > +1 ) is trivially satisfied for positive ≡ − l i z i i 2 1 i x 1 y y y x − L ′ ) and can be defined for any even m >k,L + ′ . . m k, k ( = 4, of relevance to physics, is contained amongst the faces of the object defined G →∞ >n,k m ′ 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-lineof any form the hint given cut that above. is that given For the instance as final just a result at sum 5 can over diagrams, loops, 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) ]havetodo − -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 any should with reference be the to a inside the way of of interior the 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 divide to 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 challenges theories. 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 ects can be ff ]. This perhaps indicates the 42 , whose “volume” gives us the scat- n,kL ]. We have given a formulation for planar A 19 – 17 – 26 – erent formulation of classical mechanics — in terms ff point for making the transition to quantum mechanics ff ] and pursued in [ 15 , 14 erent starting points leading to the same physics was somewhat mysterious ff 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 necessitating theory, locality the and introduction unitarity of the familiar redundancies gram put forward in [ N Hamiltonians, Lagrangians or gauge redundancies,no no mention path of integrals “cuts”, or “factorization Feynman 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 deeper unitarity are underlying theory. as a natural deformation, via the path integral. physics where space-time anddon’t perhaps appear, even then the even usualneglected in formulation and the of the physics quantum limit reduces mechanics where to perfectly non-perturbative local gravitational and e unitary quantum field 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 action is must principle not match formulation a to is central some thuslaws, but much and derived closer notion. gives to a quantum The better mechanics than jumping Newton’s o hidden in the structuredeterministic, of there classical is a mechanics completely itself.of di the While principle Newton’s of lawsthese least are very action manifestly di — whichto is classical not physicists, manifestly deterministic. but today The existence we of know why the least action formulation exists: the mechanics will obviouslyrectly be attacking a the tall quantum-gravitationalconfronted. order. questions But where The there theradical is most relevant another step obvious issues strategy taken avenue that must incal for 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 only times,division 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 for seems and by an to thus quantum extension be the 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 44 (1 × , G k 4 ] gives hope that the necessary understanding 19 – 27 – )[ ]. Remarkably, this extremely natural mathematical free of the need for any triangulation. We do not yet ,n ]) “little” 19 Ω 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 Ω 27 n,k,L k, n Ω ( 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 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 Ω 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 have an analog of thefor notion of “dual amplituhedron”,expression and should also exist. no integral Onthe representation loop a integrand related at note, any while fixed loop we 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 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 find co-ordinatesdata 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 yet of 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 space, is agive while direct 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 first, rich but cell the structure.gluing key is together The to (“amalgamating” realize task [ that of positive the matrices “big” from Grassmannian smaller can ones be [ obtained by JHEP10(2014)030 ]. ]. 58 69 =4 – , erent: N 56 ff 68 ], so it is = 4 SYM, 61 N ]. A particularly 55 – erent cyclic orderings 52 ff ], the connection between on-shell 19 erent BCFW representation of tree am- ff – 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 [ 60 U T , more 59 = 4 SYM amplitudes which does not have a usual space-time/quantum ers from theory to theory. Furthermore, on-shell BCFW representations ff cient of the log = 4 SYM amplitudes are Yangian invariant, a fact that is invisible in the N ffi 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 but rather equivalence with di physics all between 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 conventional facts in string string perturbation theory, which theory.SYM Indeed is just the 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 the and inprovides the planar the a identities limit planar they strong have case, impetus no geometric to while 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 di of scattering amplitudes are alsoand widely at available — the at very loop leastprogress level for from for other planar gravitational gauge points tree theories, of amplitudesleading view (where to [ the there amplituhedron has was beenplitudes, the much myriad with of recent equivalences di guaranteed by remarkable rational function identities relating diagrams and the Grassmannian isthe valid building-up for any of theorythree-particle more in 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 di the the measure basic on the Grassmannian determining the 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 As connection we iswhile noted, with the the the coe positive four-particle geometrycusp amplitude problem 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) ord Che- 56 ff (2011) 058 ]. 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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 EdwardJake area Witten, Bourjaily, over Andrew without the Hodges, whom past andported this many especially by work Freddy years the would Cachazo 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 Cli Drummond, Huang, Bob Jared Sasha MacPherson, Kaplan, Juan Goncharov, GregoryAlex Maldacena, Korchemsky, Song Postnikov, David Lionel He, Amit Mason, Sever, Jo- David Dave McGady, Skinner, Jan 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. Phys. ]. , in preparation. IN ]. 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