JHEP08(2019)042 = N , Springer July 28, 2019 June 13, 2019 August 8, 2019 : : : c Accepted Received Published and Matteo Parisi [email protected] , b Published for SISSA by https://doi.org/10.1007/JHEP08(2019)042 Tomasz Lukowski Tomasz a,b [email protected] , at, ¨ ¨ unchen, Germany . 3 is then the image of the positive Grassmannian via a map determined Livia Ferro, a n,k 1905.04216 The Authors. M c Scattering Amplitudes, Supersymmetric Gauge Theory
In this paper we define a new object, the momentum amplituhedron, which , [email protected] E-mail: [email protected] School of Physics, Astronomy andUniversity Mathematics, of Hertfordshire, Hatfield, Hertfordshire, AL10 9AB, U.K. Mathematical Institute, University ofAndrew Oxford, Wiles Building, Radcliffe ObservatoryWoodstock Road, Quarter, Oxford, OX2 6GG, U.K. Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universit Theresienstraße 37, 80333 M b c a Open Access Article funded by SCOAP logarithmic singularities on the boundaries of this geometry. Keywords: ArXiv ePrint: 4 super Yang-Mills theoryordinary in amplituhedron, we spinor introduce helicity bosonizedexternal kinematical space. spinor data, helicity and restrict variables Inspired them toamplituhedron to by a represent particular the our positive region. construction Theby of momentum such the kinematics. The scattering amplitudes are extracted from the canonical form with Abstract: is the long sought-after positive geometry for tree-level scattering amplitudes in David Damgaard, The momentum amplituhedron JHEP08(2019)042 18 ]. In such formula- ], and the union of the 2 6 , ], which nowadays pro- [ 1 7 [ ]. However, the cells contributing 5 , 4 amplituhedron -point amplitudes for all helicity sectors 19 n 8 positive geometries = 2 i k – 1 – ]. From this point of view, the aforementioned 4 3 Y pq h 2 ) = 10 17 18 16 Xpq 12 2 1 amplitude 6 MHV amplitudes 10 = 4 super Yang-Mills (SYM) theory. It originated from the observation that the N 3.2 NMHV 2.2 The momentum2.3 amplituhedron Momentum amplituhedron volume form 3.1 MHV/ 2.1 The ordinary amplituhedron mannian. Nevertheless, viatwistor a variables, map they defined assemble by inmannian a through a such positive convex-like map object. is matrix a ofcell geometric The space, images bosonized image the provides momentum of a the particularfirst positive triangulation. example Grass- of The a amplituhedron vast becamevide family eventually a of the geometric the description so-called for various quantities in theoretical physics: see, for instance, the physical properties ofresidues amplitudes are [ associatedpositive with Grassmannian. positroid cells, TheCachazo-Feng-Witten proper which (BCFW) combination are recursion of particular relations cellsto [ subvarieties a is inside particular selected amplitude the by are seemingly using not the related Britto- to each other inside the positive Grass- tree-level amplitudes and loop-level integrands of can be computed usingtion, amplitudes integrals can over be thewhich extracted selects Grassmannian a from space particular a [ sumthe Grassmannian of interrelation integral residues. between over the Building a rich upon suitable combinatorial this contour structure idea, of novel studies positive revealed Grassmannians and 1 Introduction In the past decade anar new, geometric picture has emerged for scattering amplitudes in pla- B Proof of the relation ( C Momentum conservation D Positive Mandelstam variables for 4 Conclusions and outlook A Orthogonal complements 3 Examples Contents 1 Introduction 2 The definition JHEP08(2019)042 . , 0 3 k 2 ] and 4+ 6 , as well i ˜ λ ,..., = 1 and A i , λ ˜ ]. In particular, it Λ satisfy particular A i ]. Additionally, fur- Z 14 12 = 4 SYM. Therefore it N ], and positive geometries in by reviewing the formulation 9 ], where it was suggested that 2 12 ]. The first states that the amplituhe- 15 hedron is more naturally suited to describe as the image of the positive Grassmannian – 2 – amplitu ˜ Λ will enforce positivity of Mandelstam variables. Then, ], the cosmological polytope [ 8 ). In order to achieve our goal, we will first introduce its bosonized i ˜ η momentum amplituhedron , ]. We will focus here on two of them, which will be relevant for our i , η ). By assuming that the external kinematic data Λ and 15 i i , ˜ λ 7 ˜ can be described on the space of bosonized supertwistors Λ , ]. , i i λ 0 =4) 11 ] through the Roiban-Spradlin-Volovich (RSV) equations [ , m ( n,k 13 10 ). The first attempt in this direction was made in [ i A The paper is structured as follows. We start in section Nevertheless, despite the name, the ˜ λ , i λ In the past fewplituhedron years there [ has beenconstruction a of lot the of momentum amplituhedron: progressthe the on description original different based definition on descriptions introduced the of in signdron [ the flips presented am- in [ technical details of our construction. 2 The definition 2.1 The ordinaryWe amplituhedron start by recalling the construction of the amplituhedron in momentum twistor space. defining the momentum amplituhedron, i.e.helicity the variables. positive Afterwards, we geometry show inmomentum how the amplituhedron to bozonized and find how spinor the to logarithmic extractconsists differential the form of scattering on amplitudes the examples fromWe which it. end Section show the in paper detail with Conclusions how and to Outlook, use and the a construction few appendices from containing section more through a map determined byindeed this a positive positive geometry, external whose canonical data. logarithmicamplitudes We differential in form will spinor encodes show scattering helicity that variables. such space is of the original amplituhedron in the bosonized momentum twistor space. We proceed by perspace ( version: (Λ positivity conditions, we will reproducether the constraints entangling sign Λ flips and postulatedwe in will [ define the sheet [ was conjectured that theas space the should additional assumption have that properthis the paper sign planar we flips Mandelstam will for variablesand show should both provides that be the a positive. proper suitable In expressions positive for geometry the with amplitude such when characteristics written exists in the non-chiral su- Amplitude/Wilson loop duality which islimits present the only possibility in of planar findingsymmetric positive models geometries and for beyond scatteringdescription the amplitudes directly in in planar the less sector. ordinary super- twistor( space It or, is even better, then in desirable thethe spinor to amplituhedron helicity find space in a momentum geometric space should be the image of the twistor-string world- CFT [ the dual Wilson looptwistor space. rather In than particular, the thealization employment of amplitude of this these itself, geometry variables to being restricts scattering the defined amplitudes possible in gener- in other the models, since momentum it is based on the the kinematic associahedron [ JHEP08(2019)042 , 0 ) 0 ), ˙ m a i n,k ( ˜ µ (2.5) (2.3) (2.4) (2.1) (2.2) Ω , ]. a i 7 , . . . k . The by e.g. λ σ 0 ) have been = 1 m ( n,k ) such that ) which can 0 ) α , A , n m , )-dimensional m ) 0 ( n,k R i k m + Ω ( · 0 χ with logarithmic 0 . + Y,Z k α R ) ( , k σ 0 G φ ω m k , n 0 4 is the R-symmetry = 4 SYM. The geo- ( 0 σ k = volume function in is found by evaluating k G α ( α i 0 and then summing over ) N . In particular, the vol- ξ m + 0 ) m ,..., · Z ( n,k log G m 0 ( n,k = 4, and in the following we Y Ω k , ∈ ) can be found e.g. in [ d A , 0 = 1 0 ) m ) ) ∧ m ( n,k αi R m c Ω . ... + i σ , one can define a ( 0 α n, m, k MHV amplitude. The components of ∧ = ( 0 ω 0 ) do not overlap and cover the ampli- Y ) k ∗ , k m 0 ) ( n,k m Z describe a positive geometry, as defined k Z d A ( 0 0 of dimension ) Y,Z k , which has logarithmic singularities on all G ( (Φ m } σ 2 ( n,k ,C via the function Φ σ ∈T α → ) Ω ...Y σ ∆ σ – 3 – X ) m { 1 ∆ ω -particle N log ]. The volume form Y is defined as the image of the map , n n h + 3 = 0 Y = 0 ) 0 k 0 d 0 ) =1 ( m ) is positive, i.e. all its ordered maximal minors are k T ( n,k Y , k α m ∧ . We start by demanding that the matrix of bosonized + 0 ( n,k MHV tree-level amplitude in , n A ) 0 k volume form 0 G m = k ( Ω k : 0 ) G Z m + Y,Z ( n,k ∈ ( Φ σ 1 Ω m A i ( α Z + αi log M c , called the Y 0 ) ∈ d = m together with the form ) ]. One can, for example, triangulate the amplituhedron encodes the N ( n,k A α 0 ∈T ) A i Ω 18 σ Y m X Z – ( n,k 0 ∆ =4) ) are the canonical positive coordinates parametrizing the cell ∆ ) is the positive Grassmannian, i.e. the space of all positive matrices modulo A m = ( n,k 16 = ( , n , 0 0 ) Ω are auxiliary Grassmann-odd parameters and 7 Y,Z k Z ( m ( ( n,k 2, and the bosonized version of the fermionic components σ i α R + , Ω α φ ) transformations. For each amplituhedron G 0 ]. Throughout the years, various methods to find the volume form An alternative way to find the volume form is to introduce a k = 1 7 ˙ a where explicit expressions for various values of parameters ( defined by The result of the push-forwardbe is a written logarithmic as differential form on the push-forward of theall canonical positroid forms cells in the triangulation proposed [ finding a collection of positroidthe cells images of thesetuhedron. cells To through each positroid thesingularities cell function on one all Φ can its associate boundaries, a see canonical [ form boundaries (of all dimensions)ume of form the amplituhedron space metric space in [ given by Here, GL( differential form will allow any values forvariables the label positive. Then the amplituhedron bosonized supertwistors include the bosonica, part of the momentumwhere supertwistors ( index. As already exploredof in bosonized the variables beyond literature, the there case exists relevant for a physics, straightforward generalization which specify the kinematic data for the JHEP08(2019)042 n ). × m = 2 ,...,n (2.8) (2.6) k . The + =1 m i 1 j 0 , where , } z i ∗ as differ- 2 i , k z 0 0 z ) Y { . However, A k i m ( − ( n,k Y Z = G 2 j Ω αi z c Y 1 and the integral i z sign flips. (2.7) c i 0 . It can be accom- X k := − z i A α ] states that the positive planar Mandelstam vari- ij Y h 2, and define the brackets , . . . , m . , 12 0 ˜ λ ] k 2 = 1 + = 1 j ) parametrized by 1 m ˙ a has exactly = 2, we consider a configuration a j m δ , satisfying particular sign-flip or [ , } a i λ by the differential with respect to . a, a z 0 i m + i z =1 i 0 k γ z , 2 Y α n } Y j ˙ , k a 1 1 0 h j ˜ 1) λ k h , ( p a − + G 0 λ i { ,..., ≤ . Let us also define z 2 1 j i ˜ λ X . . . k
[1 , k = = 2 ˙ + 2, where A i A i 0 − ˜ Λ Λ k ,..., k ˜ λ = k -particle tree-level amplitudes in [13] ) and write any function on the superspace as a differential form on its bosonic ) = ( , n ˙ a ˜ λ ˜ 3. N 2, together with the Grassmann odd parameters a, , − auxiliary Grassmann-odd parameters [12] N, n Correct sign flips: let{ the list ( Positive planar Mandelstam variables: k MHV = 1 Let us introduce 2( As described in details in [ 2 Notice that • • = 4 SYM can be written as a differential form in spinor helicity space. The starting point − 1 ˙ a k parameters. We will now repeat this construction for theand spinor 2 helicity variables. helicity variables as The degree of thisis differential a form similar in situationwhere ( to the the amplitude one can whichthe be we bosonic thought have of part encountered as ofintroduce for a a the the differential bosonized momentum momentum form momentum twistors, twistor of twistor superspace. degree space 4 by Equivalently, introducing it auxiliary was Grassmann-odd possible to Moreover, there is a naturalindices way to ( associate the R-symmetrypart. indices ( It amounts to the replacement us remark that in this space the supercharges take the form: are positive. N is the non-chiral superspacea, which is parametrizedamplitude by is spinor a helicity function variables, on In this paper weitive define geometry, the which space we ofN bosonized call spinor the helicity variablestivity and conditions a on related the pos- With bosonized additional variables, assumptions, we we will will recover also proper guarantee that sign the flips planar for Mandelstam variables JHEP08(2019)042 , ˜ Λ A ↔ (2.16) (2.17) (2.18) (2.13) (2.14) (2.15) , has rank ) is well de- , n Y ˜ Λ) , +2 (Λ k ( ) to the bosonized ) associates a pair ). The orthogonal , M as the image of the . , n 2.1 k, n 2 ∈ +2 ( +2 n,k k + 2) . + k − − ˙ k − A n + 2) in the following way n G k M is positive does not imply +2 n ˜ A i +2 Λ k k − , ˙ k − ⊥ A i ˜ Λ Λ: if we exchange Λ A i − ˜ } ∈ Λ n ... ( Λ ˙ ... αi k, n ˙ c A 2 ˜ 2 ... Λ, which will guarantee positive ⊥ αi M . 2 k, n { − ˜ A i c Λ 2 . One can show that ˙ 2 A i ∈ Λ n − ˙ -dimensional space. We denote their = ) C A 1 = ( 1 + 2) and the map Φ ˜ and Λ are positive and proceed in an Λ n 1 n ⊥ ˜ A i Λ 1 G C k ( A ˙ 1 ˜ α ⊥ Λ A i Λ G ˜ × Λ ˜ − Λ +2 . These matrices describe linear subspaces × k +2 − ˜ k Λ = positive ˙ n k, n A ) and + 2) ⊥ ... + 2) − ...A 2 – 6 – ˜ , n Λ positive ˙ 2 2 A n k, k , A 1 ( ( ) ˙ 1 ,Y − A k, k ˙ A G ( G A i , n k , the space looks the same. This corresponds to the ˜ -transformation, corresponding to a change of basis of ( G Λ k → = matrix matrix Λ ] = ˙ +2 ∈ αi M kinematic data ) − i GL c k ) ( +2 ∈ n ) through the map +2 k − = k k, n ⊥ ˙ ( n − ↔ A ˙ ˜ ( α Y,Y k, n n + . . . i ˜ ( +2, respectively, inside an Y k ˜ Λ’s we have 2 G M k + i : ˜ 1 Λ) as the = 4 SYM. In the following we will, however, need to break this . . . i G ∈ i , [ 2 ˜ Λ) i , N 1 A n is the orthogonal complement of i (Λ h Λ } Φ +2 and k ⊥ αi c ... − , we restrict the allowed external data to be positive in the following sense: { n A 2 = Λ A 1 ⊥ Λ C ), therefore it is an element of k Having defined the positive region we are ready to adapt the map ( Until now there is manifest symmetry between Λ and − Λ = n where ( fined. After imposing additional assumptions on Λ and positive Grassmannian which to each element of theof positive Grassmannian Grassmannian elements ( that the matrix Λ isone positive. can On notice the thatwill contrary, the using have matrix the both encoding discussion positive the from and the orthogonal negative appendix minors. complement of aspinor positive helicity matrix space. We define the momentum amplituhedron Alternatively, we could assume thatanalogous the way. matrices We emphasize that the fact that the matrix Λ together with exchanging parity invariance of symmetry by choosing one of two possibletwo possibilities descriptions. available These in two the choices conjecture correspondpositive for to region the the sign-flip condition. In order to define the Similarly for the space of orthogonal complements as Λ complements are defined up tothe a corresponding subspaces. Additionally, webosonized define variables. two On types of the brackets space on of the Λ’s space we of define variables. We introduce the matrices and refer to the pair (Λ of dimension In the next step we define a positive region on the space of bosonized spinor helicity JHEP08(2019)042 ) ) ) is is a n 2.20 2.20 ) and ] with (2.21) (2.22) (2.23) (2.24) (2.19) (2.20) ˜ Λ) , 6 2.24 ) in the is a pos- (Λ 2.23 , k 2 n,k + 2) is 2 − M k k n . − ) through the map + 2) and therefore k , as we will see later, ) = ( k, n ) = 2 k, n − ˜ ( k ˜ Y , N − + ]. In the context of the ˙ a i − n G MHV tree-level scattering ˜ λ 15 N, ( . k, n n 2 . and ˜ λ G + 2): − ] − → 2 in the sequence ( k ) through the map Φ Y k × ˙ ij n a i + 2( = 0 [ ( − , − , see [ ˙ . a i k k, n G ˜ k Λ ( → Y } i} · ] + 2) + ] ˜ × n Λ n k, n ⊥ 1 · G 1 ˜ Y Y ˜ − ˜ ⊥ Y ij Y k, k h [ [ -particle N ( ˜ + 2) n Y ]. Let us first remind the reader that one n ( G + 2)) = 2 G 12 a i k k, k ,..., ,..., ( i × . We defined here the orthogonal complements − Λ 4, and that we find the correct pattern of sign G – 7 – · 13 13] C + 2). One can think about the condition ( ˜ − ⊥ Y Y , , [ + 2) k, n h , n λ Y , a i , k i i − λ (2 ij 12] k, k n 12 is the definition of the ordinary amplituhedron [ n ( ( G =1 ˜ → Y i X Y [ ˜ G G Y a i { ∈ 4. {h = i → h ⊥ ˙ − Λ a a · ˜ Y n ) for Y ij P ⊥ h Y ). This corresponds to the condition ( ) reduces to the usual momentum conservation. Equation ( 2.18 2.24 + 2)) + dim( 2.20 + 2) and ]. It is easy to see that the number of sign flips in the sequence ( = 4 SYM. k k, k ( 12 − N G , n (2 dim( is lower dimensional. Indeed, the momentum amplituhedron lives in the following G The second check we would like to perform is to confirm that this geometry satisfies In order to confirm this claim, we start by checking that the momentum amplituhedron ˜ ∈ Λ) , in the sequence ( since the formula in ( ⊥ (Λ We want to showk that the number ofconjecture in sign [ flips equals k Therefore, we are interested in the following sequences of brackets: and can reduce the geometrythe in kinematic the configuration bosonized in spacemomentum the to amplituhedron, the direction the purely projection of bosonic results a in part fixed the by reduction: projecting and the condition ( implies that the image ofco-dimension the four positive surface Grassmannian inside thehas space the correct dimension 2 the correct sign flip conditions, postulated in [ as being equivalent to theamplituhedron momentum space. conservation but Indeed, written ifthen directly we we in project find the through momentum a fixed For a proof of thisY statement see appendix We notice, however, that the image ofΦ the positive Grassmannian co-dimension four surface inside itive geometry and its volumeamplitude form in encodes the has the expected dimension,flips. namely Let 2 us first observe that the dimension of planar Mandelstam variables, we claim that the momentum amplituhedron JHEP08(2019)042 . k ) is of a (2.29) (2.30) (2.31) (2.25) (2.26) (2.27) (2.28) ˜ Λ) we de- , 2, with 2.25 p (Λ − + k ) correspond ...,i ) by → +1 2, as required. , k k 2 i,i 2.30 − s − k k ( . To do this we first ) to the amplituhedron G ) is n,k . 2.8 ∈ = 2 and . . ] M ,j , . . . , k . , ). 2.23 2 k m X 4 j ˙ α 1 ˜ Λ. These are defined by c = 1 − 2.30 ,i , ˜ ˙ Y j 1 α [ } . We start by classifying possible − i ) , k + 1] = 0 2 ) are positive. Therefore ( ˙ n 2 α j n,k . The boundaries ( 1 c 1 2 ˜ , . . . , n − Y , 2 X − M ˜ 2.25 i Y i i ( Y j − k [ ˙ h k α p = 2 and + in ( Y ij i with logarithmic singularities on all bound- h ,..., ≤ , . . . , k Y 2 C ...X 1 n,k – 8 – X
g Ω n,k ˜ Y . . . k Ω in Λ M = , . . . c +2) 1 ˙ ∗ ˜ , . . . , n k a λ k ∗ 2 n i ˜ φ − ( Y ˜ 2 λ, Y n ˙ α ( = 1 c δ ... 1 , i . . . d i k ˙ αi ) 1 ˙ ) on 1 a =1 ˙ − c ˙ α A i Y ˜ α n φ n + 2) multiplied by a function. Then, the c P · ˜ Λ + 1) ( d k k 4 k ˙ on reference subspaces d α ˙ n,k αi δ Z , as an integral over a matrix space c ... ω ) and therefore there is no need to triangulate − ˜ 2 = 0 for Y k – 10 – ˙ . . . k ! α − , n − i ) Z 1 n,k n a k ˙ ˙ α (2 A ) k, n ˙ g − α and k ) + ˜ n )(23 + 1 k Y k ( − − ( − G Y − n × n ( ) ) = n . . . dφ n ) ( k k ( × Y ii · +2) . . . k 1 a g ) 2 h − k G k ( n 0 ( δ dφ is top-dimensional and therefore can be written in terms of − . . . i (12 1 × n (det ( 2 k =1 Z P i
d ˙ Y = α 4 1 ) is the canonical measure on the space of i d p = ( ( × Z + 2) n,k 4 ∗ ∧ is compatible with the embedding of n,k δ ω = Y ∗ ω n,k = ˜ k, k Y ( , n,k Ω ∗ G Y tree n,k )Ω MHV amplitudes A P ( 4 ) comes from the localization of δ p ( 4 δ : they are all of the form 2 Finally, in analogy with the ordinary amplituhedron, we can introduce an integral This choice of n, 3 )-transformation encoding the ambiguity of defining an orthogonal complement. The sion of the positivethe Grassmannian amplituhedron, it ispositroid enough cell. to take ItM the is image an of easy the task Grassmannian to top-dimensional find all boundaries of the momentum amplituhedron 3.1 MHV/ We now move toMHV study amplitudes. examples Already ofform. in momentum this amplituhedra, The case dimension starting the of with volume the function MHV momentum takes and amplituhedron a new and interesting where the brackets in the denominator are minors of the matrix 3 Examples where we additionally needk to integrate overintegration measure the matrix where will show how extracting the amplitude works in practicerepresentation in of a the few volume examples. function Ω obtaining the measure on procedure to extract theamplituhedron, amplitude i.e. from we the localize volume the form Indeed, the form JHEP08(2019)042 ’s us- α (3.9) (3.5) (3.6) (3.7) (3.8) (3.1) (3.2) (3.3) (3.4) 2 , tree 4 A ]. i i , as well as 12 14 13 Y Y . h h i 2 , . . . , n − . . n dlog i i : Y D = 1 j 2 ∧ . 14 13 d i α ] i i i 2 Y Y Y h h ˜ h Y 34 13 ) is therefore: i 2 Y Y + 1 , = h h ... 12 ˜ ˜ λ 4 , Y d i , i 2.17 Y 2 ][ 1 h . 0 for all 1 , α Y ˜ Y dlog i [23] 23] 2 Y i i h 2 ˜ ) λ 2 d . > ). A particular choice results in ˜ Y we find the following representation ∧ q Y 2 34 13 (˜ 3, see appendix Y 2 i i ˜ ! 4 ˜ Y d Y Y Y [12] [ 3 ih δ 12][ − h h 2.18 λ 4 23 13 1 ) Y d α + 1] i ˜ q Y dlog Y α Y Y [ 41] ( ih [1234] = − 2 h h i 23 4 1 2 ∧ ˜ d 3 Y h i ˜ δ 0 1 Y Y ii λ 23 2 Y i 2 1 , α h , . . . , n i ,[ i Y dlog 34][ i 2 α α 2 i i 12 Y d – 11 – 1234 ih n ˜ h ∧ Y n, h 1 0 h 1 = 1 + 1 23 13 ) + 1 i i i 12
p p Y Y Y M ( , i ) are related to each other by momentum conserva- ) and gives the following manifestly parity symmet- h 41 h h Y 23][ 12 13 4 [1234] 1 = h 2 ˜ δ Y Y i Y Y Y i, i 3.5 Y = ... h h h C and h ih = ) only in terms of 2.35 = 4) using the positive parameters i 2 , 2 ’s from equations ( 12][ , 34 2 . . . n 23 4 i , α α 2 ˜ (2 Y , Y 2.18 [ 1 Ω tree 4 i i Y + h ). This formula agrees with the result found in [ ih ) and ( = dlog dlog A ih G = 12 13 , . . . , n 1234 j 23 2.9 2 h 12 Y Y 3.3 , 1 α h h 4 Y = 1 Y − =2 ^ h i ih n Ω i and Λ: = dlog 1 12 = = Y α Y 4 =1 2 ^ h j n, = = Ω 0 for all 2 , 4 > are defined in ( ). Independently of the chosen representation for the volume form, the volume Ω p ˜ ) to get q + q, 2.20 ...,i 2.37 This calculation can be easily generalized to any MHV amplitude. A particular repre- Let us start by considering the simplest case, i.e. the four-point MHV amplitude. We +1 i,i sentation for the volume form reads ing ( where function can be evaluatedric using answer: ( unique up to momentum conservation. Finally, we can extract the amplitude It is easy totion check ( that ( If instead we solve equationsfor ( the volume form The push-forward of the Grassmannian top form through ( There are various ways todepending find only on inside the momentum amplituhedron S parametrize the top cell of in this section will make these boundaries manifest. One can also show that, for all points JHEP08(2019)042 6) , and , (3 } (3.15) (3.11) (3.12) (3.13) (3.14) (3.10) + ) = 0. G . Let us are easily ijk 2 , . . . , n 2 − − , . Finally, the (2) n,n , which results = 1 λ (612) = 0 n,n ˆ 3] ) vanishes. The A cannot be injec- , , 64] ˆ 1 56] 64] Y i ˜ ˜ ˜ ˜ M ), it can be easily Y Y Y Y ˜ ijk Λ) [ [ [ [ , 3 , (561) 6 (Λ 3.8 = = . 0 for Ω 2 4 8 ˜ is eight-dimensional and Λ + > (456) = 0 i i . , 3 p , 6 2 3 12 13 + , i (345) 6 Y Y n i M h h 1 ,...,i Ω . Moreover, for all points inside , α , α Y + +1 + Y ij h ˆ 3] h 3 i,i ] ˆ 34] ˆ 1 45] 64] (234) = 0 ˜ Λ S 3 ... { ˜ ˜ ˜ ˜ ij , Y Y Y Y [ [ [ [ (123) 6 i , . . . , n = = 4. The boundaries of Ω [12 23 3 = = 2 and ˆ ˜ = 1 Λ T − Y 3 7 = 6) for which the minor ( i
h ] ”: 5 2 ˜ Y +2 λ η . η ]. In our formulæ they are 6 3 ,i ) and shifting labels: 6 12346 | , h d (123) 6 + ˜ + 12 +1 23] ˜ λ 46] Ω λ one has i,i 3.16 ˜ i 4 + 5 ˜ Y Y | [ 3 , 3 + i 23 6 [56] + [ Y h 4 h i 23 1 ] +2 ,S η i M η ˜ Y Y 6 ” part: )(˜ 4 h → | )( i q ˜ λ
( 12345 q 6 h , 4 (˜ d by using ( i 4 δ , cancel in the sum and the form 3 ,..., 5 + 6 , δ 45] | 3 α , (123) 6 13] + 1 ˜ Y 6 → 3 ˜ Y , Ω ([ Y → ) we can further expand the Mandelstam = 1 h (123) 6 [ Ω 2 2 2 i dlog + Ω 56] , i i ]. While for the denominator it can be easily 13 ˜ 8 2.18 Y =1 ^ i 12 Y 3 , ] in h 45][ (123) 6 – 13 – [23456]) = ˜ i Y ˜ [23456]) Y 12356 Ω l [ i | i h 23 3 k , Y = 23 23 + 1] = 0 h (123) 6 12] + 56] + Y Y ˜ Ω Y ˜ h ih j Y [ 3 | , ˜ i Y ii 6 (561) [ 12 ) for extracting the amplitude, we find that the expres- + [ Y i Y 12 Ω h h i , Y [13456] + + i h 2.37 123 6 S 13 = 3 Y , [13456] + h 12346 6 (345) i h ,..., 123 Ω 13 S 46] + Y = 1 ˜ h Y ), leads to the following explicit form for the volume function , i [12456] + 3 , i + [ 6 (123) i 12 2.35 Ω Y = 0 0. This immediately implies that the two-particle Mandelstam variables are h ( i = > ) reduces to the formula found in [ 3 [12456] + 12345 , = i h 6 + 1 12 Ω 3.17 45] 3 + 1] , Y Y ii ˜ The attentive reader will notice that we have a discrepancy of signs w.r.t. [ (123) 6 h Y h 4 [ ( Ω ˜ Y ii It is clear thatthe middle the term first has and no definite last sign. term Using in ( thissuch expansion that the are canonical explicitly coordinates positive, are all however positive for positive data. Moreover, it is also easy[ to verify that for allpositive. points It inside is howeverfirst not on true for the three-particle Mandelstam variables. Let us focus appearing as poles of thediverges type logarithmically on the 15 boundaries of the momentum amplituhedron Finally, we can write the complete volume form and similarly for the volume function. One can check that the spurious divergencies, while the second bracket corresponds to the part proportional to “ sion ( seen, the numerator requires athe more first careful bracket analysis. in The the reader numerator can will convince reduce oneself to that the “ After using our procedure ( The push-forward is computed as which, using ( One can notice that the canonical variables are just an uplift of the formula found in [ JHEP08(2019)042 for 3 , , (3.24) (3.23) 6 i i -matrix M C , we find the to be positive 345 , S 123 0 S ), and subsequently (456)[13456] (123)[12346] > and R n − − ˇ [6] 234 + ⊥ i S L 6 omitted. By studying the h , n i 2). As a result, the R denote the number of particles k ˇ − [5] + + ⊥ R L,R i L n is their respective helicity. Then we 5 n k ( + G − h L L,R = 3. We encounter three different types k ∈ ˇ k [4] , n 1 ˆ ⊥ C i ), where 4 − h R – 14 – R k , n R = 6 and ˇ + [3] + k ( n ⊥ L is negative only in a very small subregion of i k G 3 ( h ∈ G 123 S One important property of the amplitudes is that they fac- R ∈ ]. To perform the amalgamation we need to start with two ) with the signs cyclically shifted by, respectively, one and ˇ C 3 [2] + C ⊥ i 3.24 (345)[12345] + (346)[12346] + (356)[12356] + (456)[12456] (125)[12456] + (235)[23456] + (135)[13456] (126)[12456] + (236)[23456] + (136)[13456] + (123)[12345] (245)[12345] + (246)[12346] + (256)[12356] 2 (124)[12456] + (234)[23456] + (134)[13456] + (123)[12356] (145)[12345] + (146)[12346] + (156)[12356] + (456)[23456] ⊥ ⊥ ⊥ ⊥ ) and : we impose the constraints on the kinematics is a positive geometry. At the moment it is unclear what is the ⊥ − h i i i i 3 L i , 3 5 6 2 ⊥ 3 4 6 , h h h h ˇ i h , n 6 [1] is positive. We can now state a sufficient condition for 1 h + + + + L h ⊥ M i k h M ( 1 123 h ] we denote the five-bracket with the index G i S ˇ ∈ ), +(456) L = (123) C 2.33 123 To illustrate how the amalgamation works in the context of the momentum ampli- S -matrices corresponding to the left and the right amplitude. we project the productdescribing to the cell whereC the factorization takes place is composed of thetuhedron, two overlapping we study it in details for geometries. For the momentum amplituhedronthe factorization the properties situation rather is come slightlythe from more the positive amalgamation involved Grassmannian of and on-shell [ diagramsplanes inside in the left andtake right their diagram, direct respectively, product, and which brings us to Factorization properties. torize into products of smalleris amplitudes reflected when planar in Mandelstam the variablesboundaries, amplituhedron vanish. then geometry the This volume in form the factorizes.is fact even In that, the more ordinary when general: amplituhedron, we the the statement approach geometry one itself of factorizes its as a cartesian product of two positive same type oftwo relations positions. ( Foramplituhedron all kinematic datageometric satisfying interpretation these of these three inequalities. conditions, the momentum curve ( for points inside where with [ remaining independent three-particle Mandelstam variables, i.e. where we have organizedthe manifestly the negative expansion terms presentpositive to in terms have making the the expansion all Mandelstam cannumerical brackets variable in negative. analysis explicitly principle Let shows dominate positive. us that over firstgeneric the remark that positive Then, a data. careful Moreover, when the kinematic data is taken to be on the moment variable to get JHEP08(2019)042 ) 0. = i → 3.24 56 (3.25) (3.26) (3.30) (3.27) (3.28) (3.29) Y 0 which 56] h ˜ Y i → . , + 1 ] , 3 j 2 i,p 1 . Y ii ¯ j I . 0 0 h 1 α j ∈ 2 2 1 i 0 3 1 0 α . 1 j α 0 0 i 2 α 2 , [ 4 j 3 4 4 2 ⊥ α 1 3 α 0 0 i 3 0 0 α α 2 α 3 M 0. Let us focus on [ α i α 4 h in the upper left corner. Notice , j 6 4 α 4 → 0 3 α α 2 , α i,p ) We finish by commenting on the ] + . α 5 I 5 3 0 can be written as 0 0 + j 6 6 α 5 matrix present in the definition of +1] ∈ 2 M 2 6 α > α j > 3 + × α α 1 i 1 + ˜ 5 j 3 2 Y ii 6 3 123 i 3 α 123 α i 6 1 , α S ( 0 1 1 i α 5 α i [ 7 α 7 456 ⊥ = 3 in details. Let us introduce the following α α 0 preserved. Indeed, the boundary with P 7 i . In particular we see that the relations ( 2 k } α 1 0 + k – 15 – i + > + p 5 5 + 3 1 i α + α − h 5 2 456 i , = 3, the conditions guaranteeing positivity for all ] α 1 + 1 0 0 3 ,i j 3 123 k 1 00 1 0 3 2 j read: α j P . It is indeed reflected in the form of the matrix defining 2 j 1 , . . . , i p = 1 0 0 0 0 0 1 1 j 2 j + 3 , and = i tree 5 P + 1 =0 2 i i A ,...,i n [ + =0 = 56 ⊥ i, i 3 +1 i Y j { 123 h 2 1 i,i | S j i | 3 1 S h , 3 := 56]=0 , 6 ,j 6 ˜ 3 Y = i [ C 0: | reduces in this limit. Finally, we consider the limit C 2 i,p 3 i 3 I j , 1 k reduces to i 2 6 → j P 4) boxes are positive and both correspond to 1 C ,j 3 , 3 × 56] tree 6 i 2 ˜ i Y A 1 i P 0. This boundary is parametrized by a seven-dimensional positroid cell for which , as expected. → The second type of boundaries we consider is [ 3 , 5 123 amplitude this boundary. Weboundary can [ see it by studying the seven-dimensional cell parametrizing the This matrix can be regarded asresponding coming to from four-point the amalgamation MHV of amplitudes.inside two positive We the matrices can (2 cor- indeed recognize that the two matrices We expect this limit to describe the case when particles 5 and 6 become collinear, and the S (123) = (456) = 0. This cell can be written in terms of positive coordinates as of amalgamations, depending on which boundary we approach. Let us start by taking 0 corresponds to the following seven-dimensional cell in the positive Grassmannian: where one can recognize thethat positive the matrix value defining of should correspond to a collinear limit with where we defined which we found in the previous section to have combination of brackets relevant in this case: We checked that, forMandelstam variables any The highlighted part correspondsM to the positive 3 Comments on positivesufficient Mandelstam conditions for variables. the kinematicdelstam data variables. which We guarantee will positivity study for the all case planar Man- JHEP08(2019)042 ]. ), i ˜ Λ)- 12 ˜ and Λ , , i ⊥ we have observed k ]. Nevertheless, this ) through this positive 12 ]. Moreover, the momen- k, n 24 ( – + = 4 SYM directly in momen- 20 G = 4 SYM one needs to factorize N encodes the tree-level amplitude N n,k ]; in particular, it obeys the sign-flip MHV all planar Mandelstam variables 12 M – 16 – that the restricted space is not empty. Extensive numerical n ] the differential forms for Yang-Mills and non-linear sigma model were 8 . Additionally, we showed that the positive kinematics, when projected to the spinor , which computes tree-level scattering amplitudes in Our paper opens various interesting avenues of investigation. The first question is As already mentioned, in the cases of MHV and After a preliminary study of the Mandelstam variables for higher ) to get a logarithmic form on the spinor helicity space, see [ q n,k ( tree n,k 4 ˜ is no underlying positive geometry. However,δ already for problem disappears when weshowed consider in the this forms paper. onbe We the anticipate possible momentum that for amplituhedron, similar, less- as but or we more non-supersymmetric complicated, theories. behaviour might positive geometries in twistor theories intum higher amplituhedron dimensions is [ formulatedvariables directly for in massless spinor theories helicityfor variables, in which investigating four positive are dimensions geometries universal (and forFor beyond). instance, non-planar, in less- This [ or opens non-supersymmetricfound. the theories. These pathway forms do not have logarithmic singularities, which would indicate that there a natural extension of tree-levelBosonizing differential those forms formulæ to in loopworth a integrands, pursuing. similar as Then suggested fashion the in asordinary [ underlying for loop-level positive tree amplituhedron. geometry level shouldcan Perhaps would bear extend the therefore our similarities most be construction with fascinating a to the question other step is theories. whether For we instance, our work could shed light on to a large number of particles tests showed that it isfor positivity actually of rather planar large. Mandelstam variables Providing is necessary an open and problem sufficient and conditions whether is left a for generalization future of work. our construction to loop amplitudes is possible. There exists are positive. Inmomentum order amplituhedron for to this bespace to a be needs positive true to in geometry,kinematical be data the other restricted remains helicity positive however further. unclear sectors, region to and us. of The to Nevertheless, the nature we guarantee (Λ checked of the algebraically up such additional constraints on the data defines a positive geometry ifthe additional volume constraints form on on kinematics are theA imposed. momentum amplituhedron Then, helicity space, satisfies the conjecture formulatedconditions in postulated [ there. In this paper we haveM introduced a novel geometrictum object, space. the To momentum accomplish amplituhedron this,for we which have defined we bosonized imposed spinor specificΛ helicity positivity variables to conditions, (Λ be i.e. we positive. demanded The the image matrices Λ of the positive Grassmannian that a similar set ofat relations the should moment be what valid also willnecessary in be conditions that and the case. what general However, is form it their of is geometric such unclear interpretation. relations, to us whether4 they also provide Conclusions and outlook JHEP08(2019)042 (A.7) (A.2) (A.3) (A.4) (A.5) (A.6) (A.1) are related ⊥ B and define the same plane B . , ⊥ ⊥ B B to be an element of the ) ) B k k B − n . , . . . , j . 1 k n . n n j , . . . , i . . . 2 ⊥ 1 ⊥ ( ) − 1 k k i ⊥ n , that in this case ( − n n } . . . . k 1 2 kn n , . ( ) ,...,j − . 1 k n ) is a scalar, independent of the minors we × . . . . b . . . b . . . b . ) ,j b − . | k k g 2 n . . . . b . . . b . . . b k ( ,...,i − − k , . . . , k k 1 n × n 2 . . . 22 ⊥ 12 ⊥ 1 − ( ,i k ) is not an involution. In the most general case . . . 12 22 k b b { 1) ⊥ n k 1 b b and b 1 | b ,...,i − – 17 – 1 } 1 T = A.3 i 1 . . . 11 21 k ,...,j b = ( 1 k b b } b dimensions. Such plane can be parametrized by an j . . . − = ( 21 ⊥ 11 ⊥ g k − b b B ⊥ n n g g b = = = , . . . , n = ˜ ⊥ B 1 ) ⊥ { B , . . . , j k B = 1 -dimensional space. We can therefore define its orthogonal B ) − j = ) transformation acting on rows of n k ⊥ n { . We motivate it by considering k } g B k − )-plane in n k in an , . . . , j , . . . , i − 1 1 j i B , . . . , j n ). We choose a patch in the Grassmannian such that ( ( 1 , j k k, n ( − : an ( n -plane G ⊥ k B matrix n , . . . , i 1 × i { ) k , with a different choice of basis. The maximal minors of matrices − ⊥ n It is important to notice that the relation ( and the basis of its orthogonal complement to be It is easy to check, by taking where consider. In our considerations wewhich will will fix a fix particular the basisGrassmannian value of the for orthogonal complement, Matrices related by a GL( B to each other ( It describes a complement In this appendix webody. set Let the us conventions consider for a orthogonal matrix complements we use in the main This work was partiallyResearch funded Foundation) by — the Projektnummern Deutsche 404358295thank Forschungsgemeinschaft “Fondazione and (DFG, A. 404362017. German Della Riccia” M.P. for would financial like support. to A Orthogonal complements Acknowledgments JHEP08(2019)042 (B.2) (B.3) (C.1) (C.2) (A.8) (A.9) (A.10) (A.11) i p q k − n ). . . . j , 1 , C.1 j ⊥ i h ] 2 -dimensional subspace +2 p q k − . k , k k 2 − − − , ⊥ n n n ) . Then, the two subspaces k ⊥ ...j − = 1 1 j C n . ˙ a 2 . i . The rule we adopt is that the , . . . , i , . . . , i − ⊆ 1 k 1 ⊥ ⊥ i i . h [ ˜ ...i Λ C B 2 , . . . , i 1 · , a, · Y p q i +2 − 1 h k i k = 1 Λ ( ⊥ − − ⊥ k · = n n ) g ˜ i = 0 Y − ) k = (B.1) i ⊥ n k ˙ a i − ,...,i ,...,i − ⊥ Y n 1 1 p q i n ˜ ,i ,...,i ,i ( Λ Λ is contained in the 2 k Y pq . This implies the following relations between 2 1 = k · · h − − ,i − +2 – 18 – B . . . j k k k k n 1) ⊥ Y ⊥ 1 · ˜ − j Y Y ( ,...,j ,...,j ,...,j . . . i ) = ⊥ k 1 1 1 . . . j j j j 1 a i − = ( 1 Y i n j h ˜ and which ones as g h ) Λ = ...j ⊥ · Xpq 1 ] = ) = ) 0 = B p q ⊥ k k ⊥ i 2 +2 2 − p q j k Y − n 2 − k k ) we need to fix − ): k , . . . , j n =1 . . . j ...i 1 . . . i i A.3 X 1 1 j , . . . , j i 1 2.28 j ( 1 i ( , . . . , j j ( [ k k 1 2 j − − − h n n k ˜ Λ are themselves orthogonal, as encoded in formula ( · X X X ⊥ <...