On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

On-shell structures of MHV amplitudes beyond the planar limit

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Citation Arkani-Hamed, Nima, Jacob L. Bourjaily, Freddy Cachazo, Alexander Postnikov, and Jaroslav Trnka. “On-Shell Structures of MHV Amplitudes Beyond the Planar Limit.” J. High Energ. Phys. 2015, no. 6 (June 2015).

As Published http://dx.doi.org/10.1007/JHEP06(2015)179

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/98234

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP06(2015)179 d Springer June 8, 2015 May 16, 2015 June 25, 2015 : : : dimensions: a n Accepted Received = 4 SYM beyond the Published 10.1007/JHEP06(2015)179 N Alexander Postnikov c doi: [email protected] , Published for SISSA by , Freddy Cachazo, b [email protected] , ), the Grassmannian of 2-planes in = 4 SYM at all loop orders can be expressed as a sum , n (2 N G [email protected] , Jacob L. Bourjaily, . a 3 ) associated with planar diagrams is just one example. The decom- e , n 1412.8475 (2 The Authors. + c Scattering Amplitudes, 1/N Expansion, Supersymmetric gauge theory

G We initiate an exploration of on-shell functions in , [email protected] Walter Burke Institute forPasadena, Theoretical Physics, CA, California U.S.A. Institute ofE-mail: Technology, [email protected] Copenhagen, Denmark Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Department of Mathematics, MassachusettsCambridge, Institute MA, of U.S.A. Technology, School of Natural Sciences, InstitutePrinceton, NJ, for U.S.A. Advanced Study, Niels Bohr International Academy and Discovery Center, b c e d a Open Access Article funded by SCOAP of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes. Keywords: ArXiv ePrint: the familiar position into Parke-Taylor factorsregions. is The simply U(1) a decoupling and “triangulation”amplitudes Kleiss-Kuijf of also (KK) relations follow these satisfied naturally extended bynon-planar the from positive MHV Parke-Taylor this amplitudes geometric in picture. These results suggest that planar limit by providingof compact, MHV combinatorial amplitudes expressions and fordifferently showing all ordered that leading Parke-Taylor they tree singularities cantended amplitudes. always notion be This of expressed is positivity assingle in understood a on-shell positive diagram in can sum terms be of of associated an with ex- many different “positive” regions, of which Abstract: Nima Arkani-Hamed, and Jaroslav Trnka On-shell structures of MHVplanar amplitudes limit beyond the JHEP06(2015)179 ]). 2 , 1 ] is based 7 , 6 9 3 2 5 2 12 8 =4 scattering amplitudes [ ] — which cannot be understood within 13 5 N – 3 – 1 – = 4 SYM motivates us to try and find a completely on-shell 13 N 1 =4 SYM in the planar limit, much less is known beyond the planar limit N ] is that planar loop-amplitude integrands can be directly represented by on-shell 8 , The Grassmannian representation of planar 3.2 Geometry of3.3 U(1) decoupling and Two KK views relations of MHV leading singularities 2.1 General leading2.2 singularities and the Classification reduction of2.3 of reduced, diagrams MHV on-shell Grassmannian functions representations and of diagrams MHV on-shell functions 3.1 Geometry of extended positivity and Parke-Taylor decompositions 7 in the Grassmannian defined by itsin 3-particle [ amplitudes. The additionaldiagrams. novelty described While weof do this not picture yet indescription know planar of whether the physics this for is general possible theories. in general, the successes on-shell processes, where the fundamentalintermediate 3-particle particles amplitudes taken are to glued be togetherOf on-shell, with eliminating course, all any on-shell reference diagrams toas “virtual have particles”. “cuts” long been which compute knownconnections to discontinuities exists of have direct loop much physical amplitudes moreparticles interpretation across are generally: most branch-cuts. naturally on-shell But computed diagrams this and in understood any in terms theory of with an massless associated structure Kuijf (KK) identities, andeach the color BCJ ordering relations separately; [ andand similarly, gravitational the amplitudes remarkable can relations not between be Yang-Mills seen in theon planar a striking limit. physical principle: all amplitudes in the theory can be directly represented as amplitudes for (despite some impressive inroadsAnd for yet four- there and are five-particleunearthed amplitudes beyond many the — indications planar limit. see that e.g. Forsatisfy there instance non-trivial [ the is relations amplitudes for an between different each even color-orderings other richer — structure such as waiting the to U(1) be decoupling and Kleiss- 1 Introduction While recent years have seen tremendous advances in our understanding of scattering 4 Concluding remarks 3 Extended positivity in the Grassmannian Contents 1 Introduction 2 General leading singularities of MHV amplitudes JHEP06(2015)179 = 4 SYM can be leading singulari- N =4 SYM, the volume- = 4 SYM beyond the N planar N ]. For 7 we show how this data directly = 4 SYM are in fact logarithmic as 2.3 N – 2 – ] have found evidence that loop integrands can be 10 , 9 ] which relate amplitudes involving differently ordered 11 ], this suggests that all MHV loop amplitudes of 9 log”-form, with manifestly logarithmic singularities in loop-momentum will lead us to an extended notion of positivity in the Grassmannian and we show that all leading singularities can be characterized by a list of d 3 = 4 SYM. We will consider an arbitrary diagram obtained by gluing black 2.2 N 2) subsets of external leg labels; and in section Because all leading singularities are on-shell diagrams, they can be computed in terms If all non-planar loop amplitude integrands in In section All of these considerations strongly motivate an investigation of non-planar on-shell The main obstacle to finding an on-shell representation of n corresponding on-shell functionspearing encode in the the loop coefficients expansionamplitudes of and beyond thereby transcendental the capture leading functions much order about ap- of the perturbation structure theory. ofof scattering an auxiliary Grassmannian integral as described in ref. [ 2 General leading singularities2.1 of MHV amplitudes General leadingLeading singularities singularities correspond and to the on-shell diagrams reductionof obtained of by loop taking diagrams successive amplitude residues integrands — putting some number of internal particles on-shell. The sets of external legs. conjectured in ref. [ expressed in terms oftree polylogarithms amplitudes with involving all differently coefficients ordered being sets color of external factors legs. times planar rically in section allow us to discover thecan be remarkable fact expressed that as the allties. positive leading These sum singularities considerations (with provide of all athe MHV coefficients geometric Kleiss-Kuijf amplitudes +1) basis (KK) of for relations the [ U(1) decoupling identities and to on-shell diagrams foressential MHV new amplitudes, features which that are occur especially for simple non-planar but amplitudes already more( reveal generally. encodes the corresponding function. Analyzing the singularities of these functions geomet- diagrams in and white 3-particle verticesfactor. together. The color-Jacobi Any relations will diagramand undoubtedly for is be the important associated hope for of with the anshell actual an on-shell diagram, amplitudes, representation and obvious of we them. color will omit However, them they in factor what out follows. of each In on- this note, we will further specialize along its boundary. Recent studiesrepresented [ in a “ space. These two factsgiven support in the purely hope on-shell that terms,at making some the the representation integrand-level. logarithmic of singularities an of integrand amplitudes can manifest be planar limit is ascattering simple, amplitudes almost becomes ambiguous kinematicalbegin beyond one: to the explore that on-shell planar diagrams limit. the beyondical the notion But interpretation planar of we limit, of which “the” can leading minimallyregion certainly integrand have singularities. in the for the phys- Any Grassmannian on-shell together with diagram a is volume associated form that with has some logarithmic singularities JHEP06(2015)179 (2.1) 2) black . n how all such 2.2 . 2.3 log’s of edge variables. Naively, d ) is finite. n, k – 3 – 2) triples of external legs — corresponding to the labels of n )) is equal to the number of independent edge weights. Unlike for the planar theory, k, n For the rest of this work, we will focus our attention on the case of reduced on- Starting from an arbitrary on-shell diagram, all but a small number of edges can always ( G 2.2 Classification of reduced,We MHV begin on-shell by showing functions that and anynaturally generically diagrams labeled non-vanishing, by reduced a MHV list on-shellthe of diagram ( (precisely) is three external legsvertices attached in (via white the vertices) diagram. to each of This the labeling ( is illustrated in the following example involving shell diagrams relevant todiagrams MHV amplitudes. can be We classified willused combinatorially, describe to and in directly show section construct how the this corresponding combinatorial function data in can section be the planar case: aBoth of non-reduced these diagram novelties can are sometimes illustrated be in the reduced following in example: inequivalent ways. of the set of reduced diagramsgraph follows to from be simple reduced dimensional if arguments. theof dimension We of may define its boundary a measurement matrixit (as is a not submanifold always possiblea to bubble transform a somewhere diagram in using the square moves diagram; and this mergers to is expose related to another important difference with descending from taking residuesthe on Grassmannian arbitrarily representation complicated makesobvious: on-shell an the diagrams. number otherwise of highly However, leading non-trivial singularities for fact any completely ( be removed (via residues settingconstraints edge on weights the to external kinematical zero) data. without encountering This any is enhanced known as reduction, and the finiteness the leading singularities couldber be of as objects complicated as atpropagators loop higher don’t diagrams, and localize with higherthe an all loops. infinite remaining the num- variables. internal At high momenta,to We can enough the and leading take loop-order, we singularities. residues however, have on the Naively, a there enough cut non-trivial are of form an these infinite in to number fully of localize objects them of this form, form on the auxiliary Grassmannian is just the product of JHEP06(2015)179 = 0. (2.4) (2.5) (2.2) (2.3) q , I n by one; therefore, − I q , n 4 counts the number of . δ −                        n ) and I + 2 + on general grounds, and so n 9 3 B 8 W I ; we want to show that 6 n n n q − (1 2 4) (1 8 9) (2 9 3) (3 6 4) (4 6 5) (6 8 7) ( , there can be no black-to-black +                        ≥ V I =0, there is one leg connected to 2 black vertices. = 3 n n B n q I 4 = 2, and n n = n ⇔ ≡ k , B = − I δ n ) n B n q 2 n denote the number of white multi-vertices. . But 3 + − internal edges by defining: q 0 . Because n V I I W and n n – 4 – n I n 4. Using this, we see that any on-shell diagram 3 + ( n = , . ≡ B 0 I I = B n n n W n n B n n − = 2 , which implies that there are no black-to-black internal V 0 ≥ ); and because 3 W I n =0 will involve n n W n 2) black vertices is labeled by three external legs according n δ + n = vertices and = 2, 3 n n + − 0 = k n 0 B = W W n W δ n n n and 2 n n , + I k + B ≡ n n =0, and hence 3 k B q n ≡ V is left invariant by collapsing all internal trees of white vertices. Let us V n I n = 2 + n B k n W n =0. The number of external legs of any trivalent graph is given by, + =2= δ B n k n Let us now show that Without loss of generality, we may suppose that every external leg attaches to the The proof that all non-vanishing leading singularities of MHV amplitudes can be la- = 2 -function constraints imposed on the external kinematics beyond overall momentum con- from which we seewe that must for have that each white multi-vertex ( internal edges. Thus every blackmulti-vertices, as vertex connects we to wanted to precisely prove. three external legs via white edges. Let us sayFrom the the number definition of of white multi-vertices is graph at a white vertex —Observe by adding that bivalent white removing vertices to a eachk external white-to-white leg if edge necessary. lowerssuppose both that this hasFor been on-shell done, functions with and kinematical let white support, vertex; no this two requires legs that can be connected to the same from which we seewith that where diagrams relatedδ to MHV amplitudesservation; have for a reducedhave diagram — an ordinary function of the kinematical data — we to the rule described above. beled in this waygram is involving fairly straightforward. We can characterize an arbitrary on-shell dia- Notice that each of the 7=( 9 particles: JHEP06(2015)179 ) k, n (2.8) (2.6) (2.7) ( G . Let us 2) black . λ ∈      in terms of                        n C Γ 4) constraints f 4)-dimensional ) is determined α (1 2 3) (1 3 4) (1 3 5) (4 6 5) (6 8 7) (6 9 8) (1 2 4) (1 8 9) (2 9 3) (3 6 4) ( n n                             C of leg labels for each , T ⊥ ∈ C ) is a (2 · ). τ α d λ ( α ) C k − log( ). Thus, the (2 n ( d × , n 2 = 4 SYM corresponds to an on-shell δ (2 with strictly logarithmic singularities. G e N λ · C ∧ · · · ∧ ) to be identical to the 2-plane C ) α 1 ( 2 α × =2) amplitude, associated with the graph, C k i k δ consisting of triplets – 5 – α log( e η T · d C =          ) 4 C × k δ (1 2 3) (2 5 6) (3 4 6) (4 5 1) (1 2 4) (2 3 4) C          ( = 2) on-shell diagram corresponding to an ordinary function Ω k Z = 0) with kinematical support will involve precisely ( = δ ], any on-shell diagram Γ of n is a top-dimensional form on Γ 7 f C which label the graph. entirely localize the 2-plane T ⊥ that can be computed in terms of an auxiliary Grassmannian ∈ C τ 2) black vertices. · is a volume form on the configuration Γ f λ C n 2) − For any leading singularity of an MHV ( We can illustrate how this labeling works with the following examples: Therefore, any MHV ( n ( × 2 subspace, and so Ω δ now describe how thesethe constraints triples directly provide us with a formula for where Ω When expressed in terms of edge-variables by boundary measurements, and Ω 2.3 Grassmannian representations ofAs MHV described on-shell in functions ref.function [ according to, Notice that because therediagram, is there no is preferred noparticular way preferred ordering to way for order to each theas order triple external completely in the legs arbitrary. the triples. of examples a And above, non-planar these so choices while should we be have viewed chosen a of the external datavertices, ( each of which iswe attached can to label exactly any three suchof external diagram the by legs ( a via set white vertices. Thus, JHEP06(2015)179 ) ∗ ~α ( (2.9) ⊥ (2.14) (2.12) (2.13) (2.11) C ’s. (The as a basis α ) matrix.) . } n b , ) by deleting ) × 2) columns in ∗ are equivalent ~α , λ 3 ~α . ( τ a λ ( ⊥ . α λ n e λ ⊥ which constitutes 3 which contributes . { · C τ i} · ) C 2 2 2 λ ∗ e λ ) λ τ − · ) and is associated with the ~α i 1 + ab λ ∗ ( τ 2) 2 h ~α T a b τ C M 2 − , ( h n i α ∈ × 2 2 2 1 C ) 2( τ × τ δ 3 .) δ can be obtained by choosing 2 λ τ 3 } δ e = 0; (2.10) η τ 2 ⊥ · h + τ i e η , λ λ a, b 2 1 , it provides us with an explicit 1 · i τ . { τ τ ) 3 T 4 ∗ 2 α τ 1 ) is found to be: , obtained from (2) transformation. (Once we have × 1 τ ~α 2 ∈ 2 τ i h ab ( τ δ 3 λ τ τ C GL 2.13 M {h a b λ 1 h Ψ 4 × / = + 2 × ≡ 2 δ } i -functions, ab δ 3 1 3 induces a Jacobian δ 3 ∗ τ τ τ e λ τ · M λ 3 – 6 – α ) i , α τ dα is completely fixed to have its ( ∗ ) from each black vertex, we find the following 1 2 h ∗ τ 2 τ 2 ~α 2 ⊥ τ τ ( τ 3 det 2.8 λ τ λ , α α C 1 dα , ih ∗ τ + λ 1 2 3 1 α τ i τ τ ) satisfying the constraints × 1 3 2 ∗ 2 Ψ = α τ τ dα δ ~α 2 ( ih τ ) to its standard form requires a “gauge-fixing” factor — a e η 2 ): ⊥ h } 7→ { · τ 1 3 ) (1)) 1 C τ -functions in this expression imposing supermomentum conser- τ . An explicit representative of ∗ does not correspond to a ‘2-plane’, but rather a (2 τ 2.8 δ λ 2.12 h ⊥ 1 ~α equal to the identity matrix. But importantly, the representative , α λ ( GL λ 2 } τ C T ∈ 4 vol( , α a, b Y τ ’s as 2-vectors in terms of a basis consisting of some pair of external 1 × { τ λ 2 . This gauge-fixing induces a Jacobian of det( = (1))” is an instruction to eliminate any one of the redundant δ dimensions, they can differ by a α } ⇒ { Γ n ) f ) constructed above is not in this gauge-fixed form; and so we must multiply GL 3 a, b ∗ τ { ; this choice of coordinates for -functions are very easy to integrate, as each encodes an instance of the unique ~α 2 } δ ( τ vol( b ⊥ / 1 ’s, the complete gauge-fixing factor Ψ of ( τ , λ λ C ( a λ { To determine the gauge-fixing factor Ψ, we need to be more explicit about how Combining the contributions to ( These Each black vertex of the diagram encodes a linear relation among the three external ’s, ’s attached to it. Thus, a vertex labeled by the triple ( its columns the final contribution to Ψ.for Thus, the having made the (arbitrary) choice of (It can be shown that Ψ is independent of the choice of λ to Ψ. Having chosen thisthe gauge complement of for matrix it on the left by the inverse of the matrix, denoted is related to the matrix to expand all the vation are not written in theiras standard 2-planes form. in Indeed, although chosen a Lorentz-frame, To bring the expressionJacobian ( — Ψ relating the two: expression for the leading singularity: Notice, however, that the from this, we see that on the support of the Because this argument isrepresentative of repeated the for plane each triple reason for introducing such afrom redundancy the here vertex is manifest.) that it makes all thethree-term singularities identity arising satisfied by generic two-dimensional vectors: λ following contribution to ( where “1 JHEP06(2015)179 ). , 2.7      (2.17) (2.15) (2.16) , i i 0    2 5 3 4 i h ) and ( i i h 0 1 3 h 0 1 4 6 2 2.6 i h i h i 0 0 0 1 3 0 0 0 5 1 6 3 i i i h h . i i h 5 1 1 2 4 1 e λ h h 0 0 · 1 2 4 6 i h h λ ), we see that i h i 0 0 2 3 1 0 0 × 2.7 3 1 5 6 2 . i i h i h δ 1 2 3 4 5 i i h e λ 3 5 2 3 3 4 e η · 1 2 3 4 5 6 h h h · 0 0 0 . λ 4 5 2 3 ) we have, λ h h i    2 4      ≡ × 2.6 5 1 × 2 ) 2 ≡ δ ih ∗ δ ) ~α i e ∗ η ( 1 3 5 · ~α τ ⊥ 2 λ ( ih 3 τ C ⊥ i 4 i 4 1 – 7 – ih C × 3 2 a b ih 3 1 τ h ⇒ δ h 2 / τ ⇒ 3 4 ×      ih ih ab Γ          2 e f τ M 2 3 , yielding the expression: 1 4 τ ≡ (1 2 3) (1 3 4) (1 3 5) ih i h Γ      T (1 2 3) (2 5 6) (3 4 6) (4 5 1) f det 1 2 ∈ 3 1          h h τ Q = : T = ∈ Γ , which is constrained by supermomentum conservation: f τ Γ e f in the particular examples discussed. Γ e f Let us see how this formula looks for some of the examples given in ( Putting everything together, we find the following expression for any on-shell diagram Following the same procedure for the 6-particle example of ( from which we see that Ψ= write For the leading singularity involving 5 particles in ( kinematical data, To simplify our expressions below, we will leave supermomentum conservation implicit and Γ labeled by triples This formula should really be viewed as defining a differential form on the space of external JHEP06(2015)179 on- . i (3.1) ). (2.18) (2.19) 69 , n ih (2 planar 76 G ) appearing ih , . α 0. The form ( 87 i ) ) ih C 3 1 > 14 τ τ i 54 ih 2 3 α 2 τ τ ih 51 ( ( i 65 ih 29 ih 2 45 ), is remarkably concise ih log i 43 ih d 43 ih 63 63 2.15 ih ih ) ) 36 ih 2 1 16 ih 25 τ τ h 46 allowing us to write: 1 3 ih 32 ih , τ τ i ( ( 2 ih 14 34 h i 46 93 ih + ih ] of the (real) Grassmannian — where i 63 ih log 62 32 d ih 16 62 29 ih – ih T ih on-shell diagram whose external legs are ih 25 56 ∈ Y 91 12 τ 51 91 ih ih h ), we find the formula: when the edge variables ih ih = 25 14 h 2.7 – 8 – 34 18 ih 3 h 3 τ + ih τ planar ) can be understood geometrically in terms of a 31 i α 41 dα ih a < b 62 ). 2 ih 2 2.15 23 τ ih τ 24 ih 2.8 α 51 dα ih 12 1 0 for ih h 1 τ 12 τ h 34 > = α h dα = ) ], given any 7 a b (1)) . This differential form is entirely fixed by its logarithmic singular- } 1 b GL , c a c vol( { T ∈ Y det τ ≡ ≡ ) C a b Ω ) is associated with the “positive part” [ As described in ref. [ As we have seen, the formula ( 2.8 ities, which can be thought to enclose some openordered region in of the the Grassmannian in clockwise ( direction aroundall the graph, the the ordered configuration minors ( top-dimensional volume form on the Grassmannian: where ( rapidly in number andsuch complexity functions with can the be expanded numbershell as of functions a external involving positive differently legs, sum ordered — itgeneralized legs. with turns notion all This out of coefficients fact positivity that 1 is for — best all with the of understood the geometry in graph of terms through the of equation auxiliary a ( Grassmannian associated 3 Extended positivity inThe the formula Grassmannian given aboveand for directly any encodes MHV allsuch leading its compact singularity, singularities. expressions: ( But although there the collection is of a possible beautiful on-shell fact functions grows that very is obscured by And finally, for the 9-particle example of ( from which we see that Ψ= JHEP06(2015)179 1)! (3.5) (3.6) (3.2) (3.3) (3.4) ] that n 7 . ) as the i 1 τ 3.3 3 ) correspond τ 1 ih 0. Because the S 3 2), which differ . τ , 1 3 2 1) , i → τ 1 n ih (1 ( 2 + τ 1 PT ··· a a τ . h 1 h i 1 = n , it was shown in ref. [ } 3 . 3) and . τ i , , we find the following expression α (1 2)(2 3) can be understood to impose the 1 2 2) i · · · h 3 λ , , τ n 3 T λ 1 , . . . , n ' 3 4 (1 , (2)) 2 C ∗ ∈ + ih , (1 n 2 3 C 1 τ τ PT τ × i · · · h , and there is no natural ordering beyond GL { 2 3 2 α α PT 7→ d 2 C 1 ih 3 4 2 τ τ − C λ vol( ih ) is the complete “tree-level” MHV scattering α 1 2 h + 1 – 9 – = τ 2 3 1 3.3 τ α ≡ 3) = 4 ih 4 α , − ) ) are positive with respect to one of the two cyclic 1 − 2 τ n 1 1 2 n , 2 h λ 2 ) localize (1 α RP 1 dα = × , . . . , n 2.8 Γ 2 PT e can always be written: 2 δ f , 3 C τ in ( (1 ∧ · · · ∧ dα (1)) ⊥ 1 2 PT 1 τ C α · dα ], and so we will refer to the on-shell differential form ( GL dα λ 1 17 = τ 2) vol( C − particles. This remarkably simple expression was first guessed by Parke dα Ω n ( n (viewed as point in Z × 2 } δ 3 ⇒ τ ) 3 , λ τ 2 2 τ τ This on-shell function is the same for all reduced, planar on-shell diagrams with this 1 , λ 1 τ τ ( λ Each black vertex can be{ seen to be inorderings correspondence of with the the three geometric constraint points. that Thus, each triple the planar limit. Howeverdifferent as “positive we will regions” now associated see,look with associated at a with single the this, on-shell black therefactor diagram. vertices are equivalent to of in Too a fact see an 3-particle many this, MHV Parke-Taylor let factor leading us involving singularity the — legs each attached to of it: which contributes a 3.1 Geometry ofGiven extended positivity the and connection Parke-Taylor decompositionsgraphs, between it is on-shell natural forms“positive” to interpretation. ask and whether Grassmannian The thedepend obvious general positivity on obstacle non-planar ordering for on-shell is the forms that planar columns also of the have the notion some matrix of positivity seems to distinct (but not independent) Parke-Taylor factors.are In two particular, cyclically-distinct for three Parke-Taylorfrom particles factors, each there other by an overall sign: ‘Parke-Taylor factor’ for the specified ordering of external legs: Notice that each Parke-Taylor factor is cyclically symmetric; and so there are ( ordering of the external legs.amplitude for Moreover, ( and Taylor in ref. [ To see that thisa follows generic from positive positivity, configuration observeto (viewed that configurations as the where an two codimension orderedconstraints neighboring one set points boundaries of collide, points of sending along for the on-shell function: clockwise around the boundarythe of canonical the volume graph form by Ω has logarithmic singularities on the boundaries of this positive region. Labeling the legs JHEP06(2015)179 0 ), 4) , > , n (3.7) (3.8) } . To (2 b (2 1 G R , c a RP G c , to bring { a . c a t        3) 7→ , a 4 c , . 2 , , ). Any (mutually con-        C 3 (1 4) τ , c b 2 , , 2 τ 3 ,PT c b , or ) associated with our on-shell , 3 1 (1 2) τ , , τ c b 3 2 , PT , τ 4        1 , τ (1 ). 2 τ . The positivity of a minor det PT 3 ⇔ a τ 1 c b        1 τ ) are cyclically ordered as points in 2 ) – 10 – 7→ τ ’s: a 3 λ c τ ⇔ 1 τ (1 3 4) (3 2 4) ) associated with the ordering of external legs around ( ) , n 4) we associate with this on-shell function changes impor- (2 , ) or ( 3 G (2 τ (1 4 3) (3 2 4) G 2 . Then, for every triple ( τ ( b 1 c b τ ) from one denoted ( < 3 ’s will be completely fixed to match the ordering of legs around the a τ possible overall orderings: (1 4 3 2) or (1 2 4 3). Because the constraints c b λ 2 τ 1 two τ However, the region of Although the choice of how each black vertex is cyclically ordered can change the final We can equivalently describe this picture as an extended positive region in the Grass- As illustrated above, cyclically orderingdered the allows legs for so that (1 4 3) and (3 2 4) are correctly or- the orientation of thelonger first fix the black cyclic vertex ordering from of (1 all 3 4) the to (1 4 3), the constraints would no (3 2 4) be cyclically orderedthe fixes poles the of overall cyclic the ordering leadingwith of singularity the legs lie columns to along ordered be the according (1 boundary 3 to 2 of 4); (1 3 the and 2 4). so positivetantly all region if of we choose to reorient either of the two black vertices. For example, by changing Notice that the constraint that (1 3 4) be cyclically ordered together with the constant that the graph and thecan resulting be on-shell seen function in must the be following example: a single Parke-Taylor factor. This For example, it isordered not consistently with hard the toordering graph see of (all all that ordered the ifboundary clockwise, every for of black example), the vertex then graph.simply of the the In cyclic a positive region this planar of case, diagram the were geometric region associated with the graph is simply means that diagram, we can choosesistent) an choice ordering of for such the orderingsand associated our thus on-shell defines ( form an has extended logarithmic positive singuarities region on in theexpression the boundaries by at of most this an region. overall sign, it can significantly alter its geometric interpretation. indicate how each blacktriple denoted vertex ( is to be ordered,mannian. will henceforth Let distinguish usthe between use columns a “little to group the projectivized rescaling” form on each column of constraint that either ( JHEP06(2015)179 ) ).) 2.8 (3.9) 3.5 larger (3.11) (3.12) (3.10) 3) , 4 , 6 , 5 , 2 3) 3) 3) , , , , 6 2 5 . (1 , , , 2 5 6 3) , , , , 4 5 6 PT 4 , , , , 2 4 4 2 , , , } . , 3 τ (1 (1 (1 6) + 3) , resulting in the Parke-Taylor (1 , , } <σ 3 4 PT PT PT 2 , , τ PT 4 5 ) — those for which all the triples . , , , <σ 5 2 ) , n 1 6) + 3) + 5) + 5) , , τ (4 1 5) n , , , , 2 6 σ , (2 2) + 5 6 3 4 : , , 4) associated with this function is , , , , , T G , 3 4 5 2 3 is cyclically ordered correctly. (1 (1 ∈ , , , , , (2 2 6 2 3 τ 4 T , , , , , , . . . , σ |∀ G (3 6 4) 6 2 4 4 PT PT 1 ) ∈ , , , , , (1 n σ τ Z ( – 11 – (1 (1 (1 (1 / n PT 4) + 3) + S PT PT PT PT PT , , ( (2 5 6) 6 2 are cyclically ordered correctly in each of the Parke- + + + , ∈ , , σ = 3 4 X T 4) = { , , } , 5 6 ∈ 2 = , , , τ 2 5 (1 2 3) Γ 3 , the region of e , , f , (4 5 1) (1 (1 , ≡ { (1 0 ). For one choice of how to order each triple, we see that the T PT PT PT weaker 2.18 (3 4 6) ). Moreover, each of these formulae connects the leading singularity − , 4) + 3) + ) for which each triple , , n 3.10 6 4 Z , , (2 5 6) / 5 2 , n , , denotes the set of cyclically-inequivalent permutations, and the sum is taken 3 6 S n , , ( 2 5 Z , , ∈ / (1 2 3) n σ (1 (1 S ≡{ T PT PT Notice that each of the choices for how to orient the triples produces a different expres- This identity is quite non-trivial. Let’s illustrate it for the 6-particle leading singularity More generally, there can be many orderings of the external legs consistent with a + sion of the formwith ( a different region withinnumber the of Grassmannian relations — satisfied providing by a Parke-Taylor geometric amplitudes basis involving for different a leg large orderings. Taylor factors above.labeled Changing the the diagram by orientationdecomposition: of the last two triples, we could have Notice that each of the triples over all written explicitly in ( compact expression becomes the sum of seven Parke-Taylor factors as follows: functions of the external kinematical data: where choice of orientation for eachmore triple. than When one this ofare is the the cyclically different case, positive ordered the partscorresponds correctly. combined of to region the will Because boundary span the determinedis by boundary cohomological the of to triples, we the the know positive combined the volume region sum form of exactly in ( positroid volume forms. Expressed in terms of than a single positive region. And we now have a geometric basis for the identity: (Here, the minus sign follows from the reorientation of a single triple according to, ( on overall ordering are JHEP06(2015)179 ) -1 (3.15) (3.13) (3.14) . , . . . , β 2) 1 . . )                          } , n, β ) ) ) ) ) ) 1 1 , n -1 -2 -1 n , . . . , n, n n β β β -2 3 n 1 2 2 , -1 -1 ...... 1 , . . . , β α α α (1 -1 β β 1 β , . . . , α -2 { n n n 1 α α PT ¡ , . . . , σ ( 1 ( ( ( ( ( } 1 , α +                          -1 β , σ ... n (1 + 1) , . . . , α 1) 1 − PT α { = ( ∈ σ X                          ) ) ) ) ) , . . . , n 2              -1 ) ) 1 ) 2 n n n β β n n ) = 1 2 1 1 -1 -2 -1 -1 ...... , n, α α α n (1 1 1 -2 – 12 – 2 n β n β n β (2 3(2 4) 4 5) (2 ( α α ( 1 ( ( ( ( ( PT              , . . . , β ].                          1 ), even a planar diagram can be decomposed into a = = 18 ) 3.8              ) , n, β 1) n -1 ] can both be given a geometric interpretation in this way. 1 . . . n , . . . , n 11 n 2 , (2 (2 3(2 4) 4 5) (2 , . . . , α (1 1              1) , α PT − ( − (1 PT × β n 1) − ( Similarly, we can derive the KK relations in their standard form by choosing different Although the U(1) decoupling identities are of course included among the KK relations, Notice that this relates a planar diagrama involving sum legs ordered of (1 Parke-Taylor factors: orientations for the black vertices of the following planar diagram: they are often presentedit in different is ways. useful And todecoupling so identities consider for can the the be sake twothe seen of cases following to clarity planar separately. follow and diagram from illustration, (with In the different equivalence their orientations of chosen most two for familiar representations one of black form, vertex): the U(1) identities and KK relationsWe [ can also understand theseresented relations as as a residue sum over theoremsindependently certain which by deleted can Andrew edges be Hodges, of graphically [ the rep- on-shell diagram, as has been observed As we have seensum in of the different Parke-Taylor example factors ( ifthe the planar black ordering; vertices are andfunction oriented the follow in equivalence geometrically a of from way these contrary thealent to fact different volume that formulae forms they for correspond on the to the same cohomologically Grassmannian. on-shell equiv- Indeed, it turns out that the U(1) decoupling 3.2 Geometry of U(1) decoupling and KK relations JHEP06(2015)179 ). ), . =6, 2.8 i n 3.10 (3.16) 24 ih 2 i 45 ih 63 ). However, results in the ih 63 =3 and 2 ih k 25 λ 3.11 46 ih ih

24 5) have a pole when 1 h 34 , λ . + 4 ih i 2 , 3 62 ) are purely logarithmic i 62 , ih 2 ih 46 , 3.3 56 6 h 52 ih 2 , i ih 25 (1 25 34 ih h h 2 i 0 by sending PT 23 h 23 h i→ 2 = λ in the denominator above would seem to 12 2 h

6) and 2 i 1 i , −−−→ λ 5 63 , i 2 5 4 h ih , 14 – 13 – 3 ih 25 , these relations (as well the BCJ relations) are , 2 k ih 51 and , ih 2 24 2 i (1 h 45 i ), makes all of the physical singularities manifest at the + 23 ih h i 63 PT 63 ih . Making use of Schouten relations, the numerator becomes: 62 ih 2.15 2 25 ih ) makes manifest the fact that all singularities are logarithmic, λ 46 ih ih 52

]), and this is one avenue towards a non-planar classification. 14 1 ih 34 3.10 h λ ih 19 + 34 i h 62 ih ). Taking the residue about 62 ih 56 ih 51 2.18 ih 25 ), and as the positive sum of Parke-Taylor factors given in equation ( ih 34 h 31 2.15 ih 23 0; but this does not correspond to any singularity of the on-shell function, and the ih ), it is likely that a complete combinatorial characterization of them should be possible. We have also given a geometric interpretation of the U(1) decoupling and KK relations Of course, the fact that all the iterated singularities of ( This distinction can be illustrated in the case of the 6-particle leading singularity 12 i→ h n, k 16 It is for instancebe clear made that planar by (as cutting in enough [ legs, anyfor non-planar Parke-Taylor on-shell factors. diagram can usually For associated general with statements about complete amplitudes; it would be interesting to We have seen that non-planarof MHV the leading planar ones singularities with canwe different always have orderings be checked of written that legs.decomposed as this Already a into for sum is differently the ordered noknow case planar on of longer general ones, the grounds and that case( new there — objects are a not appear. finite all number But of leading since objects singularities that we can can appear be for any h corresponding residue cancels between the two terms. 4 Concluding remarks involves a measure with only logarithmic singularities. is made clearthe by Parke-Taylor its expansion representation introducescombination. in spurious For terms example, boundaries both of which Parke-Taylor are factors, cancelled ( only in Ψ factorizes in the limit The ability of themarkable, numerator but Ψ follows to from correctly cancel the all fact apparent that double its poles representation is in quite the re- Grassmannian, ( expression: The presence of the doublesuggest poles non-logarithmic behavior. However, closer inspection indicates that the numerator cost of obscuring the fact thatParke-Taylor expansion all ( these singularities are strictly logarithmic;but in at contrast, the the cost of introducing spurious boundaries whichgiven join above each in of ( the positive regions. We have now described twoterms quite of different combinatorial ways data to describing representequation the MHV ( on-shell leading diagram: singularities the in Each compact formula of given these in representations makesular, different the aspects compact of expression the ( functions manifest. In partic- 3.3 Two views of MHV leading singularities JHEP06(2015)179 ], 9 , = 4 ]. log- d , N ]. ] SPIRE IN SPIRE , ][ IN (2010) 125040 ][ D 82 The Complete Four-Loop arXiv:1007.4297 [ super-Yang-Mills Theory = 4 ]. arXiv:1004.0476 [ N A Duality For The S Matrix Phys. Rev. arXiv:1106.4711 (2010) 54 , , with an interpretation in terms [ k SPIRE IN ][ ]. 205-206 (2010) 061602 New Relations for Gauge-Theory Amplitudes Perturbative Quantum Gravity as a Double The Structure of Multiloop Amplitudes in Gauge SPIRE (2012) 025006 IN 105 – 14 – ][ D 85 Five-Point Amplitudes in arXiv:0805.3993 super-Yang-Mills Theory [ ), which permits any use, distribution and reproduction in ]. = 4 ]. N Phys. Rev. Scattering Amplitudes and the Positive Grassmannian Phys. Rev. Lett. , , Nucl. Phys. Proc. Suppl. SPIRE arXiv:0907.5418 SPIRE , [ IN IN [ ][ CC-BY 4.0 (2008) 085011 This article is distributed under the terms of the Creative Commons Supergravity D 78 (2010) 020 log-form”. Since all the leading singularities can be written as a linear com- ]. d = 8 03 N SPIRE IN arXiv:1008.3327 [ JHEP arXiv:1212.5605 Phys. Rev. Copy of Gauge Theory and Gravity Theories Four-Point Amplitude in [ and Finally, it is natural to combine the results of this note with the conjecture of ref. [ N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, N. Arkani-Hamed et al., Z. Bern, J.J.M. Carrasco and H. Johansson, Z. Bern, J.J.M. Carrasco and H. Johansson, Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, J.J. Carrasco and H. Johansson, Z. Bern, J.J.M. Carrasco and H. Johansson, [6] [7] [4] [5] [1] [2] [3] Attribution License ( any medium, provided the original author(s) and source areReferences credited. by the Government ofthrough Canada the through Ministry Industry of Research Canada &and and Innovation; Ellen and by J.T. the Lee is Province supported Postdoctoralnumber of in DE-SC0011632. Scholarship part Ontario by and the by David theOpen Department of Access. Energy under grant We are grateful to Andrew Hodgessupported by and the Lauren Department Williams of foris Energy helpful under discussions. supported grant N.A.-H. number by DE-FG02-91ER40654; is a J.L.B. F.C. MOBILEX is grant supported from by the the Danish Perimeter Council Institute for for Independent Theoretical Research; Physics which is supported forms in the loopplitudes. momentum variables, This multiplied suggests by thatSYM color after at factors all integration, and loop the Parke-Taylor am- (unordered) non-planar orders Parke-Taylor are MHV factors. expressible amplitudes as in a sum of polylogarithmsAcknowledgments weighted by color and of Grassmannian geometry. that non-planar loop amplitudepressed integrands in have “ logarithmic singularitiesbination of and Parke-Taylor can factors, we be expect ex- that the integrand can be represented as explore whether they have on-shell avatars for general JHEP06(2015)179 3 . ]. 56 Compos. , SPIRE IN [ The All-Loop (1998) 70 2 (2014) 261603 J. Am. Math. Soc. (2011) 041 , ]. 01 113 Phys. Rev. Lett. , Logarithmic Singularities and SPIRE math/0609764 JHEP Singularity Structure of , IN , [ Represent. Theory SYM , Lie Theory and Geometry, In Honor of Phys. Rev. Lett. = 4 ]. , Gluon Scattering , in N n , Birkhäuser,Boston, U.S.A. . SPIRE ]. IN – 15 – [ arXiv:1412.8584 ]. , Positroid Varieties: Juggling and Geometry (1994) 531 SPIRE IN [ 123 (1989) 616 ]. ]. An Amplitude for arXiv:1111.3660 Multi-Gluon Cross-sections and Five Jet Production at Hadron [ B 312 SPIRE SPIRE ]. IN IN ]. ][ ][ Prog. Math. , Nonplanar On-shell Diagrams and Leading Singularities of Scattering SPIRE Total positivity, Grassmannians and networks SPIRE IN arXiv:1411.3889 IN Canonical bases arising from quantized enveloping algebras Total Positivity in Partial Flag Manifolds Total Positivity in Reductive Groups [ (2013) 1710 , [ Nucl. Phys. , 149 arXiv:1008.2958 arXiv:1410.0354 Amplitudes Math. (1986) 2459 Bertram Kostant Maximally Supersymmetric Amplitudes Colliders (1990) 447 Integrand For Scattering Amplitudes in[ Planar Maximally Supersymmetric Scattering Amplitudes [ B. Chen et al., A. Knutson, T. Lam and D. Speyer, S.J. Parke and T.R. Taylor, A. Hodges, private communication. G. Lusztig, G. Lusztig, A. Postnikov, R. Kleiss and H. Kuijf, G. Lusztig, Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, [9] [8] [19] [16] [17] [18] [13] [14] [15] [11] [12] [10]