JHEP12(2019)073 a Springer , October 7, 2019 : December 10, 2019 November 18, 2019 : : Received Accepted Published Andrew J. McLeod d [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP12(2019)073 Cameron Langer, basis of integrands for both theories, so that the c same [email protected] , [email protected] . Enrico Herrmann, 3 , d a,b 1909.09131 The Authors. c Extended Supersymmetry, Scattering Amplitudes, Supergravity Models

We extend the applications of prescriptive unitarity beyond the planar limit , [email protected] E-mail: [email protected] University of Copenhagen, Blegdamsvej 17,Center DK-2100, for Copenhagen the Ø, Fundamental Denmark Jefferson Laws of Physical Laboratory, Nature, Harvard Department University, ofSLAC Cambridge, Physics National MA Accelerator 02138, Laboratory, Stanford U.S.A. University,Center Stanford, for CA Quantum 94039, Mathematics U.S.A. Department and of Physics Physics, (QMAP), University of California, Davis, CA 95616, U.S.A. Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, b c d a Open Access Article funded by SCOAP ArXiv ePrint: divergences. Importantly, we use the presence or absence ofby residues inspecting at the infinite cutsplementary loop of material. the momentum theory. becomes Complete a details feature of detectable our resultsKeywords: are provided as sup- tering amplitudes in bothintegrand maximally supersymmetric basis Yang-Mills we theory construct andsingularities is gravity. that The diagonalized ensures on thetheories. a manifest matching spanning As of set ainfrared-divergent all of consequence, parts, soft-collinear non-vanishing this with singularities leading integrand hints in basis toward both naturally an splits integrand-level exponentiation into infrared-finite of and infrared Abstract: to provide local, polylogarithmic, integrand-level representations of six-particle MHV scat- Jacob L. Bourjaily, and Jaroslav Trnka Prescriptive unitarity for non-planar six-particle amplitudes at two loops JHEP12(2019)073 28 19 30 31 16 33 29 24 12 sYM 15 28 5 5 7 23 38 23 – 1 – 21 1 numerators for non-planar integrands : the residues of loop amplitudes chiral -matrices. S B.1.2 Kinematics:B.1.3 on-shell functions of Closed ordered formulae (MHV) for amplitudes all two-loop MHV leading singularities B.1.1 Color factors at two loops: diagrammatics & notation On-shell functions In recent decades, many new methods for perturbation theory have been developed, B.2 Two-loop leading singularities of MHV amplitudes in SUGRA A.2 Explicit expressions for the numerators of basis integrands B.1 Color-dressed leading singularities of MHV amplitudes in 3.2 Putting everything together: MHV amplitudes in sYM and SUGRA A.1 Representing (functions of) four-momenta 2.1 2.2 Generalized and2.3 prescriptive unitarity at A two good loops start: 3.1 Prescriptive integrands for a spanning set of leading singularities in the full amplitude.of many This known deeply obscures the symmetries and mathematicalleading to simplicity enormous progress ining our our ability understanding to of compute the scattering mathematical amplitudes structure and that enhanc- underlies quantum field theory. Scattering amplitudes constitute fundamental objectstheory, in where any they weakly coupled describe quantum the field diagrams basic provide interactions an of algorithmic and elementary broadly particles.quantities, applicable the While recipe Feynman expansion Feynman for suffers the from computation seriouslems of computational — these and most conceptual significantly, prob- theprocess, large number which of individually diagrams depend that contribute on to unphysical each (gauge) scattering degrees of freedom that cancel 1 Introduction and overview C Organization of the supplementary material B Explicit leading singularities for sYM and SUGRA 4 Conclusions and future directions A Explicit numerators for six-point integrands 3 Six-point amplitude integrands of sYM and SUGRA Contents 1 Introduction and overview 2 Principles of generalized and prescriptive unitarity JHEP12(2019)073 – ], 8 45 – 6 ], but 54 ], scatter- —a rational 17 , 16 = 4 sYM theory ], and symbols for N integrand 49 , ]— 48 40 ). ]. 1.1 34 ]. Compare this with the case ], and on and on. Much of this 44 , 31 ], on-shell recursion relations [ 43 – 5 ] (and its extension to the Yangian – of rational functions. Such integrand 22 1 38 We use the strategy outlined in [ , ], color-kinematic duality [ 1 34 basis 15 – 9 ]. – 2 – 51 , 50 ]. The basic idea underlying this method is that loop 5 nonplanar integrand’ is problematic due to the lack of global labels – 1 ]. [ the 37 – 32 ], various bootstrap methods [ ] and two loops for five particles [ 21 42 – ]) of maximally supersymmetric Yang-Mills (‘sYM’) theory in the planar , 18 41 39 , 35 generalized unitarity ] for recent attempts to remedy this problem). When we refer to the integrand in a nonplanar ], local integrands through three loops for arbitrary multiplicity and helicity [ = 8 supergravity (‘SUGRA’)—the present state of the art is only five loops for 53 27 , N 52 In this work, we construct a four-dimensional integrand basis for six-particle scattering It has historically proven difficult to construct bases of integrands large enough to Among the most important strategies for obtaining and representing amplitude inte- A key insight underpinning a number of those developments was the realization that Note that a global definition of ‘ 1 ], function-level results through seven loops for six particles [ at two loops in (nonplanar)also sYM take and advantage of SUGRA. a crucial simplification that last(see occurs [ at thissetting, multiplicity we — mean it the knowledge is of all coefficients in the decomposition ( four particles [ of planar sYM amplitudes,loops [ for which we47 now have four-particlefour-loop integrands seven-particle through amplitudes ten [ represent scattering amplitudes atprisingly high few examples loop are orderthe known or beyond arguably particle the simplest planar multiplicity. four-dimensionaland limit quantum Indeed, of field any sur- theories theory. [ Today, even for propagators on-shell — must factorizetrees). into products This of idea lower-loop becomesrepresented amplitudes in a (ultimately, powerful terms tool ofbases once any can it therefore sufficiently be is large constructed, realizedtheory studied, that and or integrated scattering loop independently process. integrands of any can particular be limit was discovered first as a symmetry of loopgrands integrands is [ integrands, as rational functions, should beresidues. constructible in In terms particular, of residues their multidimensional of loop integrands — which put some subset of Feynman that lead to easier loopphysical integrals. properties Good integrand manifest. choices canbeen additionally a In make rich important fact, sourceFor of the example, insight structure into the of thesymmetry dual-conformal nature loop [ symmetry of integrands scattering [ has amplitudes historically in various theories. the computation of perturbative scatteringof amplitudes ‘summing can be Feynman divideddifferential diagrams’ into two form to steps: on obtain that thedifficult a problem space of representative of loop loop integration. internal‘easy’ In loop and particular, momenta; the ‘hard’ vast and loop difference that in integrals difficulty of has between tackling motivated the the more development of integrand techniques geometric descriptions of on-shelling physics equations [ [ progress has been driven byTime concrete and calculations time beyond again, theshockingly reach such simple of computations results recent [ have imagination. yielded unanticipated surprises and often These developments include generalized unitarity [ JHEP12(2019)073 (1.2) (1.1) as defined . pure ]. In terms of . 58 , 4 B . Details and explicit 2 : the color-dressed on-shell . B is that it is normalized and large enough to represent both } i ⇔ } ) {I 2 N 10 f , ` 1 ` ( i {I + permutations i I – 3 – N i f denotes the leading singularity in either sYM or i , and formulae for all SUGRA leading singularities X and coefficient N 10 in the corresponding theory. (For less supersymmetric f = B.1 N in our basis involves a collection of Feynman propagators i 8 correspond to the same set of cuts — just interpreted in f , 10 i f =4 MHV I , N 6 .) A ]), we have engineered our basis of integrands to be 3.2 56 , 55 . ⇔ B.2 10 I ). It has been conjectured that MHV amplitudes in sYM are polylogarithmic 1.1 ]. That is, all basis elements have unit leading singularities on all co-dimension eight Before moving on, it is worth highlighting several key aspects of the representa- As an example, integrand Another desirable feature of our integrand basis In addition to being individually polylogarithmic (i.e. all integrands are individually 57 log’ as defined in [ d these tree amplitudes arethe fully gauge dressed group; by inreview color-factors these the built ingredients latter from below, case, structure with constants the full details of trees provided are intion simply appendix ( those of supergravity. We will This basis element isresidue theory-independent; of unit it magnitude isof associated normalized with its to exactly propagators, one have andbasis. of a to the co-dimension vanish solutions Depending eight on to onSUGRA, cutting the built the all defining from eight context, cuts the of tree the amplitudes remainder in of the the corresponding integrand theory. In the former case, definitions for each of these functionsfunctions are collected in in sYM appendix areare described defined in in with the topology diagonalized according tosuch a a spanning basis, setthe the of relevant leading coefficients ‘leading singularities needed singularities’theories, to our [ basis represent would have to amplitudes bepresented in extended; in however, we any can this always theory choose paperof the are integrands to leading simply represent singularities a we subspace have of chosen the is larger enumerated basis.) in table The spanning set residues and are expectedintegration. to Moreover, evaluate many to ofto maximal the support transcendental integrands residues in weight associateddivergences. our functions with As basis after such, soft-collinear — we divergences those expect — not this are directly representation to free defined be of well any suited infrared for integration. two different theories.permutations (We in will section be more explicit about what‘ is meant byin the [ sum over ticular, we construct a basistheories of simultaneously, loop deriving integrands ataking representation the of form (the integrand of) these amplitudes In both cases, the coefficients possible to construct a basis of local integrands that are manifestly polylogarithmic. In par- JHEP12(2019)073 , ) ]. 2 44 and m 66 ( same , O 65 . In particular, . way to upgrade a B ]; it is also known that simple 55 ]. It furthermore opens up avenues we review the basic principles of prescriptive unitarity 72 and closed formulae are provided for 2 . – dimensions in a similarly nice manner, A ]), but we outline the broad themes in . Our main results are summarized in C 67  , ], suggest the usefulness of starting from 2 54 64 2.3 64 − ]. The basis of integrands we construct here – – 59 60 , and summarize how these can be used to fix 60 – 4 – 2.1 users. The detailed organization and content of the Mathematica ]. ]); these residues at infinity cancel via global residues theorems in 76 45 ] as well as an extension of the empirical second entry conditions for five- 75 , and how the particular integrand basis needed for six-particle amplitudes in – , together with discussions on its most salient features. More complete details of 3 2.2 73 , In addition to these details, we have included with our submission as supplementary This paper is organized as follows. In section Our six-particle results represent a natural step forward in complexity, following recent We should emphasize that the representation we have derived is strictly four- 56 , material which provide fullyplain-text usable expressions definitions for of alltwo-loop each our ingredients. six-point integrand MHV We and have amplitudestional coefficient prepared functionality in needed for sYM for andsupplementary the material SUGRA, representation are and of described we in appendix have provided addi- sYM and SUGRA wassection constructed in section our results can bedefining found the basis in integrands the areall appendices. given the in required appendix Specifically, leading each singularities of of the sYM explicit and numerators SUGRA in appendix we discuss on-shell functionsthe in coefficients section for a givenThe integrand construction basis of in unitarity the basesof unitarity for this based non-planar expansion work theories of (see issection amplitudes. the somewhat beyond forthcoming the work scope of ref. [ for exploring a potential extension of55 dual conformal symmetry beyond theparticle planar limit amplitudes [ [ generalized unitarity and its refinement in the guise of a partial basis ofThus, we integrands expect whose our differentialbut result equations leave to that are generalize to in to future 4 canonical work. form [ developments at two loops for five particles [ the case for dimensionalthe regularization, coefficients must however, be wherefour-dimensional changed. the integrand basis to There find must doeslessons something be not from suitable yet enlarged for the exist dimensional any currentparticle regularization, state-of-the-art, amplitudes but in including sYM the and recent SUGRA examples [ of two-loop five global residue theorems (usingloop momentum). the fact that we have matcheddimensional. all If the the residuesof infrared at the singularities theory, finite then are nothing regulatedcorrections about by to our going the representation would to integrand need coefficients the to would Coulomb change vanish branch (as in the the massless limit). This is not starting at two loops,with amplitudes multiplicity, starting in for SUGRA sixhas have particles term-wise poles [ support at onintegrands infinity with in poles degree ref. at [ growing infinitysYM, (analogous while to the the non-vanishing representation residues of of one-loop SUGRA are indirectly matched by the and free of poles at infinity to all multiplicity and loop-order [ JHEP12(2019)073 = 8 (2.1) (2.2) N , ], and its refined form 5 = 4 sYM and – 2 N ]. 83 +1 , we outline the principles of generalized 2 L + 2.1 1 X states L = – 5 – L , where we describe the essential strategy behind = , and discuss technicalities and strategies involved in      2.3 2.2 — in section ]. Although these methods are well known, it is worthwhile : the residues of loop amplitudes 47 ]. For example, for a three-loop amplitude we could iterate the cutting      81 – . For any unitary quantum field theory, the residues on such poles are 2 =0 77 2 ) =0 on-shell functions unitarity [ 2 Q Res ` + ` . ( on-shell 2.3 ] where the loop amplitude factorizes into two lower-loop on-shell processes: On-shell functions 82 Consider, for instance, the famous unitarity cut involving just two on-shell propaga- This section is organized as follows. After briefly introducing the starring characters This happens for any theory of massless particles in four dimensions, as the on-shell three point am- prescriptive 2 also exist for massless theories in any number of dimensions [ among others. plitude cannot be corrected in perturbation theory, and is therefore fixed at tree-level. Similar statements This cutting procedure need notcorrespond stop here to — trees. further residues canprocedure be until taken we until find all vertices diagrams such as propagate along these on-shellset lines, of and states in that particularrealizes can involves that, summing be at over the exchanged. amultiplicity level scattering complete of This processes. the basic integrand, idea such becomes residues a only involve powerful lower-looptors tool or [ when lower- one The most important idea involved in unitarity-basedfield methods theory, the is only that, poles for any arisingsome local in quantum the states Feynman diagram expansion corresponddictated to putting by on-shell scattering processes, which involve only the physical states that can in this story — and prescriptive unitarity in section applying these ideas to (especiallyin MHV) section amplitudes in non-planar quantum field theories 2.1 these ideas can safelybuilding skip prescriptive to integrand bases section for non-planar theories,details. and introduce For some a notational moresuggest thorough e.g. discussion refs. [ of the ideas underlying generalized unitarity, we Our results for theSUGRA follow two-loop from the six-particle basic MHVof principles of amplitudes generalized in unitarity [ to briefly review bothrepresentation the we have basic found ingredients for and six-particle the amplitudes. philosophy guiding Readers the already prescriptive familiar with 2 Principles of generalized and prescriptive unitarity JHEP12(2019)073 (2.3) 3 ] which constructs an 52 . They are defined to residues from localizing dL ]. But this is by no means the spacetime dimensions, these on- 77 d ]), these can be computed relatively 87 – on-shell functions 85 in the adjoint representation of some Lie abc – 6 – f ) gauge theory [ = c : expansion. ]. These on-shell functions correspond to maximal- N , and simply trace over the color indices associated c bc ) a 58 /N abc , T f 4 ), in which vertices represent amplitudes and every edge [ internal on-shell propagators, or 2.2 dL ] for more details. Since all relevant building blocks entering 84 ] (as implemented in e.g. [ 7 , 6 ] for recent work). . Specifically, being interested in a theory with all adjoint states, we associate 89 , leading singularities 88 B.1 or multi-trace expansion of SU( expansion (except insofar as this expansion truncates at any fixed loop order). Finally, we should mention the special role played by a particular class of on-shell func- In the current work, we take a pragmatic view of color factors, described in detail in One aspect that is still quite poorly understood is the role of color factors in the Pictures such as those in ( c c It is worth contrasting this situation with the very interesting work of ref. [ 3 /N /N In the first ofbeing these, the cut, dashed the line momentum signals flowing that in through addition that to edgeinteresting three and should internal propagators novel bewithin representation set the of multi-trace to framework non-planar zero of multiloop the (making 1 integrands the for sYM, but very specifically dimensions include: tions called codimension residues of loopshell amplitude functions either integrands. involve In fewer propagators but cuttingof additional these Jacobian poles in (composite four residues). dimensions at Examples one and two loops relevant to MHV amplitudes in four straightforward to expand these1 graphs into more familiaronly representations gauge — group such we as might1 the consider, and we are not limited to any particular order in the appendix to each particlealgebra a with color structure matrix constants with ( all internal on-shellgraphs particles. built out This of results structure in constants. color Once factors a that particular can gauge be group encoded is as specified, it is subtlety) in the definitionany representation of of on-shell any gauge functionsand group); for however, the the amplitudes kinematic interplay with between functionshowever, these non-trivial [ color that color factors they (in decorate is still largely uncharted territory (see, the on-shell functions describedcomputed above in are any tree-level quantum amplitudes,recursion field relations they theory. [ can In in particular,easily principle using to be modern high tools multiplicity. such as BCFW classification of on-shell functions. To be clear: there is absolutely no obstruction (or even represents an on-shell physicalbe state, the product are of called can the propagate amplitudes at at the each edges.space vertex, This summed of state over the sum all internal involves theamplitudes). an particles internal integration states (often over See that the localized on-shell ref. by phase [ momentum conservation at the vertex JHEP12(2019)073 ) 2.4 , are (2.5) (2.4) ]. Dual 98 , integrands ], considerably 38 , -gon at infinity if 91 p 35 , of loop integrands: set of basis integrands. 34 loops is to simply write , ]. This opens up certain 5 [ basis L good 95 – ,  91 , ) 2 i 14 /` ]). (1 . 94 is said to scale like a ]). For leading singularities supporting O 0 i I I 90 i c 1 + of integrands at  i } p 0 i X ) – 7 – alone suffice for the representation of six-particle 2 i 1 ` {I = ( A = dual-conformal invariance I MHV — the latter involving at most three particles. By →∞ lim i ` ]. We take the former view, in which the basis of integrands ( 97 leading singularities , 96 singularities. In the planar limit of sYM, there exists a simple guiding principle for constructing One way to construct such a basis It turns out that A fair amount is known about the leading singularities of sYM beyond the planar limit, leading sufficiently large integrandconformal bases: invariance is closelycounting’ of related the theory: to a what planar is loop integrand colloquially described as the ‘power- and in generalrepresent the the amplitudes problem in of aseparate given constructing and theory arguably is a more an important interesting basis questionLet and is of important how us to one. briefly find integrands discuss a However, a just the role large of enough these questions to in the context of sYM. on loop momenta [ is simply an independent set of rational functions. down all Feynman diagrams in adependent given parts theory, of and consider each. thesize (These space of integrands spanned the will by space the rarely loop- required be to independent.) represent Interestingly, amplitudes the in different theories varies considerably, Such an expansion existscase, it both should before really be and viewed after as a loop statement integration about a — cohomology where, basis for in rational the forms latter 2.2 Generalized andOnce prescriptive it unitarity is at realized two thatrational loops differential perturbative forms scattering of amplitudes, internal and viewedbe external as expanded momenta, loop into it any becomes sufficiently clear that large they but can otherwise arbitrary for these processes existsthis is that not is true, even purely init polylogarithmic. the is MHV worth sectors Beyond of briefly sYM sixfreedom describing and in particles, how SUGRA. these To integrand however, understand basis bases this canbe subtlety, can be be fixed according constructed, to how cuts, the and degrees when of these cuts can be taken to have a representation ingeometric terms interpretations of and theabout powerful Grassmannian how computational [ these tools. functionsless are is Although understood classified much beyond for this is MHV case known amplitudes (see, however, in [ amplitudes sYM in [ sYM and SUGRA for the simple reason that a complete basis of integrands MHV helicity configurations, it isat not the hard vertices to are see either thatconvention, MHV the these or only are helicity colored amplitudes in allowed our figures by blueas and well white as vertices, in SUGRA. respectively. Like any other massless theory in four dimensions, on-shell functions number of constraints equal to four) (see e.g. [ JHEP12(2019)073 (2.6) (2.7) . It is i ) graph- q ]. But let 2.5 54 , 2 ) 3 q ], there does exist -gon integrands’ at + p 54 c ( 2 ) 2 q -gon power-counting’ to be + p        b ( 2 ) 1 -loop planar integrands with box ) as contact terms. q L loop-independent momenta + 2.6 a , when these loop momenta are defined any = ( of loop momenta — no preferred choices of n , ,...,L – 8 – ), for = 1 2.6 i routing with numerator        ]. 27 internal loop momenta, or for how these loop momenta should flow through L Notice that our definition of power-counting is entirely graph-theoretic (and thus in- At two loops, a basis of ‘triangle power-counting’ integrands may be defined as the Beyond the planar limit, however, no simple definition of ‘power-counting’ exists owing As some of the authors will describe in a forthcoming work [ straightforward to generalize this examplegraphs to that construct contain spaces either of of numerators the for scalar arbitrary triangles independent ( of how thedimensionally internal agnostic. loop momenta The are spacetime routed dimension through appears the only graph) in as the well question as of how scales like the first scalar integral in ( (or better) at infinite loop momentum. For example, the integrand amplitudes in sYM and SUGRA. space of allcounting loop integrands integrands that scale like one of the following scalar triangle power- this, we may definethe a space of basis all ofinfinity integrands non-planar (or that integrands better). behave with likebeyond The ‘ a precise the specified definitions needs set of ofus of these this qualitatively ‘scalar bases describe work; the are we space defer somewhat of a involved two-loop and integrands more required go thorough to discussion represent six-particle to ref. [ a graph-theoretic definition ofwell-defined subspace power-counting of that Feynman allowsrepresent integrands us amplitudes (which to may in or a enumeratetheoretically: may given a not theory). rigorously a be The pair largeterms essential of enough behave idea to loop graph-isomorphically is as integrands to all have reinterpret their ( the loop same momenta ‘power-counting’ are if taken to their infinity. leading With results up to ten loops [ to the fact thatorigins there for the is no preferred the graph. by the planar dual ofplanar the sYM integrand’s are graph dual conformally of invariant, propagators. and(among manifest other It dual important conformal is conditions) invariance known requires that thatis integrands amplitudes have not in 4-gon terribly (‘box’) power-counting. difficultpower-counting, and It to to construct use this the basis space to of represent amplitudes all in planar sYM which led to for every loop momentum indexed by JHEP12(2019)073 Considering are listed just the rank red . power-counting? triangle – 9 – ] and the ranks of the degrees of freedom for their ) integrand topologies consistent with ‘triangle power- 54 d . ]. Thus, at least some integrands with triangle power- 1 59 ] amplitudes in sYM are free of poles at infinity through 59 modulo contact-term degrees of freedom are listed the total rank of numerators, and in power-counting integrands to form our basis, as such integrands almost SUGRA, and we know that, starting at six points, amplitudes in SUGRA blue and power-counting basis to represent six-point amplitudes in sYM — as, one na¨ıvely, . A complete list of irreducible (in 4 triangle sYM The first reason is that we would like to be able to have a basis large enough to represent Although the above motivation is a good one, it may not actually be the most important Before we describe the much harder problem of finding (reasonably) good representa- Specializing to four dimensions, we can compute the ranks of all these vector spaces support residues at infinity [ to us. Indeed, we strongly suspect that it will be much easier to represent amplitudes in both do counting are required in our basis (and completeness requires that we consider them all). that we knowthree that loops MHV (and [ chosen suspect this toalways hold have support to on allthis poles loops), choice. at it infinity. There may are be two surprising main reasons that why we we have have made tives of each required integrand,triangle it is worth briefly discussingwould expect why we the have theory chosen to to be use expressible a in termsWhy of integrands have with we box chosen power-counting. our basis to have degrees of freedom may be spanned by their contactof terms. numerators and enumeratecomplete the irreducible list degrees of ofnumerators freedom two-loop is that integrand enumerated must in topologies be table fixed. with irreducible The triangle power-counting counting’ as defined andnumerators. described In in ref.spanned [ by numerators many independent degrees of freedom exist for these numerators — and how many of these Table 1 JHEP12(2019)073 - d (2.8) ]); and 47 log forms d spanning set propagators are L —residues of maximal =4) results in an elliptic d ]. 45 Let us suppose that some ini- . ) would be determined by a linear 2.4 -theory, but it also represents a compo- 4 in ( log forms). A natural question is: when i ϕ d c leading singularities log form. Note that integrals in d – 10 – = : ]. How is its coefficient matched? One can show that no db better one. The virtues of such bases were described in I 102 exist, as cutting all 7 propagators (in much could easily be large sums of on-shell functions with (in general, db I i c ]), then all the coefficients ] (which is true for all theories in dimensional regularization via ] to be associated with polylogarithmic functions of maximal weight 99 100 14 are trivially just the cuts used to define the basis. ] for recent progress on integrating i c 101 ], where they were called ‘prescriptive’ due to the fact that in such a basis the 47 Whenever there exists a spanning set of A minimal set of cuts sufficient to determine all coefficients is called a The two main problems with constructing a basis of integrands with box power- Such an integrand is requirednent to amplitude represent in scalar planar sYMleading [ singularities of can this happen?unfortunately, is: Can no. cuts always Considerdouble-box: for be example chosen any to integrand basis be which leading includes singularities? the scalar The answer, coefficients codimension — then theare amplitude conjectured has [ a (see e.g. [ algebraic) prefactors arising from the Jacobians of the basisof integrands cuts. on those Given cuts. anyresulting spanning in set (typically) of a ref. cuts, [ it is possible to diagonalize the integrand basis, Choosing bases andtial finding integrand (perhaps coefficients farcut-constructible from [ optimal) basisdimensional cuts of [ integrands isalgebra problem known. of matchingbasis, Provided cuts the the coefficients with theory field is theory. Depending on the initial choice of pentagon integrands, but these pentagons arenant related constraints due in to the four existence dimensions.requires of Gram Thus, choosing determi- the only construction five ofcourse of a be the (not made, but over-)complete six lead basis pentagons to additional to undesirable appear. complexity in the These expansion types coefficients. of choices can of counting in four dimensions arerequired that: to form first, a integrands basis withities — more than and they such than have integrands 4 degrees typically of havesecond, many freedom such more (leading an leading to singular- many integrand delicate basisall choices, integrands is see of topologically e.g. a [ over-complete. given topology By are this independent. we mean At one that loop, not for example, there are six like boxes (at leastreason until for this a is better a bit definitionof) subtle and of box involves two power-counting a unfortunate in aspects ‘box fourone of power-counting’ dimensions. (any loop existing basis where, Luckily, definition both in exists). retrospect, problemsof are The these triangles problems already into were familiar the the at ‘chiral motivation box’ behind expansion the introduction described in ref. [ sYM in terms of integrands with triangle power-counting than using those which behave JHEP12(2019)073 ]. ] for 117 (2.10) ], these 117 ; (2.11) – 46 ) in bases, 112 2.8 , ]). Associated 103 111 – ) (2.9) ) have MHV support; 104 x ( Q ⊃ 2.10 = 2 y ’: x where , ]); in the latter case, the integral is known y dx and 117 – 11 – Z , , = 116 , 103 ) in any basis large enough to represent amplitudes in sYM 2.10 ⊃ ] over the remaining degree of freedom ‘ 103 , 102 ) is an irreducible quartic. have cuts which are non-vanishing for even MHV amplitudes. We must therefore x ( do Q In recent years, a number of researchers have started exploring amplitudes and individ- To be clear, the existence of non-polylogarithmic integrals (those without leading sin- of integrands. that include terms such as(or ( any lessthat, supersymmetric beyond theory, the for planarlast that limit for matter). and at which two From we loops this can or argument, maintain higher, we manifest six particle expect polylogarithmicity amplitudes together are with the a local basis of rigidity (in theto sense be of finite, refs. and [ Careful to readers involve will an notice integralas that over such, neither a these of cuts K3 the should surfacehave maximal vanish irremovable cuts — for support of all a within MHV ( Calabi-Yau integrands amplitudes. two-fold [ Nevertheless, these cut surfaces for seven and eightmaximal particles, co-dimension respectively. residue of While the integral the is first an case elliptic integral, is suggesting infrared some degree divergent, the high enough loop order,theories, massive these particles, issues or arise sufficientlyloop earlier many cut external beyond surfaces legs. the associated planar with In the limit. massless integrands To see this, consider the two of points along the cut surface. ual integrals outside the worldto of these polylogarithms cases (see, are for integrandstous example, feature without [ of leading almost singularities, all which amplitudes turn in out almost all to quantum be field a theories ubiqui- [ and gularities) does not in anynor way from prevent matching us their from coefficients. includingdegrees Indeed, integrands of as such motivated freedom as (and can ( exploited) simply in be ref. fixed [ by matching the integrand at any sufficient number integral [ JHEP12(2019)073 (2.12) (2.13) integrand . ) 12 some αβs + 23 1 ) shows a transcendental leading singularities. The βs + 2.13 any 13 αs ( , αβ ]. = 119 , ). But we can in fact do better: knowing (or ) does not have , 118 2 – 12 – 2.13 2.12 β p = b and numerators for non-planar integrands 1 α p 7−→ = chiral a integrand basis. One well-motivated strategy would be to choose an good Rather, our guiding principle will be to choose integrands that are naturally engineered Given the above considerations, it is clear that much thought must go into the choice Again, we could fix the coefficient of such an integral by matching field theory anywhere There is one final subtlety regarding the integrand basis construction that should be ficients in SUGRA). In suchdirectly, a and basis, all relatively the fewnot others leading the singularities would approach would follow that be from we matched describe completeness here. or residueto theorems. match as This many is physical singularities as possible. In this basis, the composite leading basis, and a integrand basis such that asome maximal amplitudes number of of interest). integrandsto have This a vanishing strategy coefficients very is (for different indeedone representation a could of reasonable choose amplitudes one, as thanintegrands but many would what it necessarily defining we would have contours derive lead vanishing as coefficients here. in possible sYM For to (and example, involve fewer poles vanishing coef- at infinity; such the question of the correctthese choice can of be spanning tuned cuts. to There expose are different2.3 many aspects choices of available, field and theory. A goodAs start: described above, there is a substantial difference between constructing along the codimension-six cut surfaceexpecting) of that ( MHV amplitudessuch in integrands sYM from should our be basis. maximal weight, we mayof simply a cull ‘good’ integrand basis. In the case of six-particle MHV amplitudes, there still remains Upon taking any furthercut-pathways, residue signaling that we the find integral ( aexistence double of pole. such Indeed, doubleweight this poles drop, arises is confirming in a the all known signal result possible that [ the integral ( is a simple example of ancut integrand without parameterized any by leading singularities. Consider the collinear considered. Even for integrands without ellipticit curves may or not K3 be surfaces possible inFor their to six cut find structure, particles a at spanning set two of loops, leading the singularities scalar to integral fix their coefficients. JHEP12(2019)073 (2.17) (2.15) (2.16) (2.14) 2 matrices × box integral, In order to fa- i α scalar δ require any particu- a, ) 2 = c , by infrared-divergent · not . 1 56 c p

( 2 . ) + q , ··· A . directly γ 2 + β d a ) 2 ( 2 A.1 c d, d  b q 1 · ) but with ‘triangle’ power counting, 2 1 b = b 2 4 ( a p span 2.15 β α ) ∈ + ): 2 ˙ ) α ⇔ a ) — those directly responsible for the soft- ` · α µ ( 1 σ n a 2.3 – 13 – c, c ( h ]). Further details — including the bracket’s sym- = = : 23 122 with  p – 2 ,c + 2 . Elsewhere, this bracket has been called a ‘Dirac trace’, 1 120 d ,c γβ 2 · )  , c ` · , ( 2 ˙ b, b γγ 2 ). Momentum conservation can always be solved for example 2 n b j (e.g. [ a ,b 2 ˙ p = γ  1 ˙ a α 1 , but it will be extremely useful for us to ,b  + p ` ˙ 2 α ··· : α 1 ,a  + = ... into the Pauli matrices 1 a a a = a : µ +  = p : ]’ i represent the momenta flowing through the propagators of the graph. Of β p α ··· ) [ 2 = ( : + a j · 1 ··· i a ]), which leads to a numerator of the form p a, b, c, d integrands tend to vanish on these cuts. 47 If we want to define an integrand like that of ( Before we describe how these integrands may be constructed, it will be useful to define we follow the definitionref. of [ power-counting discussed in the previous section (and that of where by declaring that lar solution to momentumloop). conservation (especially In for particular, non-planarnumerators we integrands of have beyond our used one basis this of graph-theoretic integrands described description in of appendix all loop-dependent where course, momentum conservation requires (in all-incoming conventions) that Constructing chiral numerators formiliarize box ourselves with integrands the at notationfeatures, one-loop. introduced above let and to us betterthe first understand simplest its look example salient relevant at towhich some our (prior discussion to examples is any of the normalization) ‘two-mass-easy’ we chiral denote integrands as at one loop. Perhaps where ( denoted ‘tr metries and degenerations — are discussed in appendix one recurring bit of notation‘bracket’ for built these out numerators. of Specifically, traces we(by of find (pairs dotting it of) any useful four-momenta to expressed define in a terms of 2 collinear divergences ofintegrals. loop Upon amplitudes diagonalizing — theof are integrand integrands basis, matched to this bedivergences tends infrared are to render finite associated a (a withchiral maximal feature parity-even subset contours we in consider loop-momentum quite space; valuable). as Soft-collinear such, singularities such as that on the left of ( JHEP12(2019)073 / ∈ 2) 2 ) and − q ∗ 1 (2.20) (2.22) (2.21) (2.18) (2.19) d + a contact a . 1 e λ
. . |

2 is a solution for 56 2 ) ) [14] p . In terms of the ∗ 1 | ∗ 2 i q on the cut a is the solution for a n [4 + ∗ 1 − a d a ) would correspond to ( , = ( :  q 2 ∗ 2 14 ) : to have support on one a s to normalize the residues ∗ 1 2.15 a 2 = , span ) ⇔ exclusively −  4

1 chiral ' a p 1 ). From this, one learns that the , ( ,

µ 4 µ 2 + 14 e ) · s , ` q 1 ` 2 p b, c, ` = + :   = ( 2 c : spacetime dimensions. = ( ( n = : µ  d q ` 2 14 span n s ' and span

are mutually parallel (and thus would support – 14 – 2 ' ) and , , } q are mutually parallel (and thus would support non- 2 4

2 ) i } ) + ∗ 2 , 4 4 q 2 | a a i ) ( ∗ 2 56 + = 0. In four dimensions, there are precisely two solutions  − c, d, p 14 q p a b { | h 2 a ( c, d, p d ( −  1 + 2) for (integer) { q . These elements of the numerator basis are called 14 λ a = span d } ( s and 2 = 14 c ), these numerators could be written as : and = } span : s ∗ 1 1234 1 1 } = a ' 1 n = , p 2 2.14 

b . 2 4 ⇔ 123 ) ∗ 2 a, b, p = q . Notice that all the contact terms vanish on solutions to the so-called a { to express each component of a, b, p , p

2 1 { + 2 µ a , , b, c, MHV amplitudes at those vertices), while the solution e ` can be any spacetime vector. In particular, as the propagators are simple a , p , d 1 ( 0 2   q ’s of q ’s of λ , c = e λ 2 ∈ { 1 ) to be unit in magnitude. Notice that the requirement of chirality would be , b n span q 2 . After removing the contact terms from the space of numerators, there are ( a 2.20 We’d like to choose our remaining numerators to be In the language above, we are interested in finding two numerators of the form ( A very natural subspace of numerators for the integral (  where we haveon included ( the normalization left invariant by the addition‘brackets’ defined of in any ( combination of contact terms to vanish on the cut solution or the other, but not both. This is easy to arrange: take, for example, which the non-vanishing which the vanishing MHV amplitudes atof those helicity vertices). flow ensure It that will MHV be amplitudes useful have later support that considerations The cuts above have been decorated in order to emphasize that span ‘quadruple cut’ to the cut equations, taking terms degrees of freedomremaining remaining. numerators in four Let dimensions. us now describe a very natural choice for the two where we have ignored overalldinate factors vectors of mass-dimension, andrank used of a spanning such numerators set is of ( coor- translations of each other, Moreover, one canpolynomial show in that this space is spanned by a Lorentz-invariant degree-two Notice that JHEP12(2019)073 ,  4 (3.1) ], the (2.23) , b, c, 45 1  . (Notice also and ∗ 2 a  3 = 8 supergravity. , 2 N . , b, . 1  ]; it can be expressed in 57 numerator involving one or , , , 4 1 c ], a closed formula was guessed e e λ λ λ 8 any ∝ ∝ ]). For example, in ref. [ ∝ c b b e e λ λ λ -point MHV amplitude integrand in n : 123 In ref. [ , = 8 a on cuts for which the so-decorated corners + only = 4 super Yang-Mills and ’s of all momenta involved at this vertex parallel. – 15 – λ N X apropagator integral The second term inbasis, the as it numerator is vanishes directlynaturally on proportional wonder the to the ‘defining the role inverse cut’ it— propagator for plays. in this It this element turns case,coefficient. of out diagonalization that the That with this is, respect term this to is second fixed an term via integrand in diagonalization in the our integrand basis can with be vanishing understood as representing However, all of thesoft integrals internal momentum in which our isautomatically basis parity ensures associated even; that thus, with the these chirality sub-topologies of the involve numerators a above These integrands include as sub-topologies many of the other graphs appearing in table JHEP12(2019)073 .) is A.1 exactly signs between in isolation = 8 SUGRA are N relative ), by having integrands of the terms in the sum. 3.6 involve an ordering of the ), color-decomposing every ] and the references therein.) 3.7 signs One rather subtle aspect of any 125 A nice feature of our result is that = 4 sYM and ). Specifically, we verified that, upon necessarily N ]). Specifically, although we have chosen 3.2 124 ) involves the for both – 20 – 3.7 , respectively. Further details, including machine- N i f B.2 and . B.1 C ) has a minus sign in front of it? After all, we could have defined sign ambiguity. This final ambiguity was fixed by comparing the 3.4 , explicit numerators are given for all 37 integrands (with non-vanishing in ( ). Beyond merely fixing the final overall sign ambiguity, this comparison 9 overall I 3.2 A.2 ) to the planar limit in equation ( ), and converting each of these integrands to dual-momentum coordinates c 3.7 /N Having fixed all relative signs, and checked that all poles at infinity vanish, there Even if we concede that the sign of any term is arbitrary, the In appendix regions can become quite intricatecent for discussion higher-loop, in large-multiplicity the amplitudes. context (For of a QCD re- amplitudes, see e.g. [ Infrared structure: finiteness andmany of divergences. the integrands inble our for basis infrared vanish explicitly divergences indivergences as the a can soft/collinear result arise regions of in responsi- ther choosing massless soft chiral or scattering numerators. collinear amplitudes In to where general, external infrared loop momenta. momenta Enumerating all become potentially ei- infrared-divergent on-shell function oforder sYM, in extracting 1 the(valid coefficient in of the the planarmatches single-trace limit), that of term that ( (at theamounts leading resulting to rational a loop-momentum very integrand strong test of our result’s consistency. check of our result, but also fully determines thestill relative remains signs one between allexpression terms. ( summing all integrands together with their permutations in ( some residues at infinity tonormalized vanish manifestly on — these such contours as in thatof our of our basis ( basis (with coefficient integrands zero)at (so and that infinity diagonalizing they whose the all residues rest Checking individually vanish that vanish)—there in all are poles a many at very other infinity non-trivial vanish poles for way sYM via therefore Cauchy’s not residue only provides theorem. a very serious the numerator of the integral’s coefficient with a differentsign sign. should But be — understood more as intrinsically — conventional, if how its is coefficient’s thisterms sign are determined fully determined at by all? tation, Cauchy’s the residue theorem ‘Global (or Residue its Theorem’ multi-dimensional (GRT) manifes- [ Explicit checks and eliminatingrepresentation sign of ambiguities. an amplitudeIt such could as be ( arguednot that well the defined overall (or signpropagators arbitrary), of being as any cut on-shell particular via residues on-shell a function Jacobian. As such, the reader may wonder for example, why coefficients) in our basis.The (The notation coefficients used (leading is singularities) describedgiven explicitly in in some appendices detail inreadable appendix formulae, can be foundare in documented the in supplementary appendix material attached to this work, and momenta to soft momenta.permutation (Remember invariant, that and thus amplitudes the innever ordering sYM matter.) of and Thus, external SUGRA we legs aregenuinely must attached distinct always sum to cut over the configurations. all same re-labelings vertex of the external legs that result in the white/blue decorations of the vertices (the parity of the cut solution), and also map soft JHEP12(2019)073 — vanish } 27 I , 26 I , 23 I , 22 = 4 sYM, such an analysis {I , as well as those integrands divergence upon integration. N } 2 12 I / ,..., 1 {I – 21 – ], however, similar analyses can be extended beyond the pla- ], we can probe potentially dangerous infrared regions at the 46 125 ], this hints toward the exponentiation of infrared divergences at 46 MHV cut solutions, respectively. Taking both residues thus restricts us to divergence upon integration. Similarly, when two adjacent three-particle ver- / ]. In our setup here, it is easy to confirm that the integrands associated with 57 The six-point amplitude is the largest multiplicity for which both sYM and SUGRA can This partitioning of our basis into explicitly infrared-finite and infrared-divergent sub- Diagram numerators can potentially cancel the composite poles described above. a specific, physical leading singularity. be expanded into the sameexactly power-counting basis. the Notably, same the form resultsfunctions. in and both only Moreover, theories differ take the by choice the of interpretation chiral of numerators the for corresponding individual on-shell basis elements nicely 4 Conclusions and futureIn directions this paper weMHV have amplitudes constructed in the maximally four-dimensionalintegrand supersymmetric integrands Yang-Mills representation for theory includes two-loop and six-point tained supergravity. only these Our pure, results via unit-leading-singularity prescriptive integrals. unitarity, where each basis We integrand ob- directly matches spaces seems important, rather thansector a of mere sYM curiosity. in Likethe seen ref. integrand-level. at [ Could two we loops use in thesefor representations the remainder to planar build functions manifestly or finite ratio integrands functions? We leave this question to future work. not specifically engineered to match ain soft-collinear divergence, all leading infrared-divergentis regions. constructed This to happensintegrands vanish because in on each the the basis of are MHV their infrared numerators cut divergent by solutions construction. for these vertices. The rest of the Whenever this happens, we determine thatsponding the infrared resulting singularity. integrand Prime does examples note.g. of have in the this corre- [ effect arenon-composite the chiral leading integrals singularities defined — namely, the intersection of theseexpects two a cut 1 surfaces, wheretices all (each three involving momenta a are singlelonger external collinear any leg) and momentum are flow one both throughus made the we collinear propagator are in connected in this to a way, both soft-collinear there limit, vertices. is and no This should tells expect a 1 nar limit without muchlinked effort. to In integrands the that language haveresidues support of can unitarity on be cuts, certain takenleg. infrared composite whenever three-particle divergences residues. vertices are Since Collinear involve one composit aagators of single factorizes external the the (massless) three secondthe legs propagator MHV is into and already a massless, pair cutting of simple either poles of — the corresponding internal to prop- integration crucially depends oncancels whether the the divergence. numerator ofregions In performed each lieu in diagram e.g. of softens [ integrand the or level fully power-counting via analyses unitarity cuts.has of been the In performed the infrared-divergent in context ref. of [ planar Furthermore, the question whether or not there will be an infrared singularity after loop JHEP12(2019)073 ], ], as 126 , 75 – 59 73 – 22 – dimensions in order to exploit the loop-integration  2 − ], and should therefore be expressible in the basis of triangle power-counting 56 , ], for example. Integrating the amplitudes as we have represented them, however, 55 76 Another natural path forward would be to integrate the six-point MHV amplitudes we We expect that all amplitude integrands in sYM will be free of poles at infinity to all Going forward, there are several natural pathways towards a better understanding of (CF18-0641) (AJM). The research of J.T.of and C.L. Energy is supported grant in partsupported DE-SC0009999 by U.S. by and Department the U.S. by Department the of Energy funds under of contract DE-AC02-76SF00515. University of California. E.H. is This work was performedNational in Science part Foundation grant at PHY-1607611, the andSciences the Aspen Harvard and Center Center of for Applications. Mathematical Physics,(No. which This 757978) is project supported and has by aFoundation been grant (341344, supported from LA) by (JLB); the an and ERC Villum supported Starting Fonden Grant by (JLB); a by Carlsberg a Postdoctoral grant Fellowship from the Simons may require upgrading them totechnology available 4 in dimensional regularization.ential equations We hope may help that guide the this framework line of of differ- furtherAcknowledgments investigation. have found. This willas not that be hinted easy, by but thewell may as ‘non-planar lead dual-conformal being to symmetry’ a thetries described natural discovery [ in of testing refs. new ground [ simplicities for such questions about the restrictions on symbol en- amplitudes in supergravity require largerhavior integrand at bases infinity in for general largeat due multiplicity. to infinity Moreover, their the appears worse behavior to be- ofand be supergravity suggests in amplitudes that conflict additional with (andintegrands cut-constructibility novel) in conditions in supergravity. must four be dimensions found [ to construct amplitude — allowing us tolimit: probe the 6-point the ratio firstresults function. to intrinsically In higher infrared-finite a multiplicity similar function or vein, loop beyond it orders. is the natural planar toorders try [ and generalize these integrands (with some definition of ‘triangle power-counting’). In contrast, it is known that of their associated physical singularities. perturbation theory beyond the planaramplitudes, limit. integrands Although will we have requirebasis shown that we non-polylogarithmic even have integrands constructed for MHV is beyond sufficient 6-points, for the 6-point NMHV amplitudes in sYM and SUGRA delineates integrals that are infrared finite and divergent, matching the infrared properties JHEP12(2019)073 and and were (A.1) (A.2) (A.3) (A.4) (A.5)
i 1 b a a | : , λ a = λ a λ has less than ˙ α of spinor-helicity by α i p ˙ α = det a α : indices of the local e λ δ p i ) α 2 ab λ c — as an ‘empty’ trace : h . · L,R . 1 = 2 for (not necessarily mass- c ! i q ˙ ( α 3 1 2 . α b p a α

λ ip  p ··· ˙ 2 = 2 α c γ − − α

. ··· β q 2 matrices constructed using the 1 0 ) ˙ , α a p  p 2 2 y × γβ ··· e λ , b q 

· 2 x 3 of a massless external state labelled 2 , 1 is null, the matrix = ˙ p γγ 2 : b a ip = a

i ( a p p 2 identity matrix.) ˙ 1 b  γ β ··· + + | ˙ a α α q 0 ×  1   ) | : p ˙ 2 p 2 q, q α a ’s. ); we choose to introduce new notation here q a =  α 1

· 2 γ a 1  = ) correspond to SU(2) , c = 2 a – 23 –  1 ( = : ˙ ,c α h 1 β A.1 α µ ··· ). An example of where this more familiar notation α ,c = , σ : ) · , µ 2 y ·  , , , . . . , c p a in ( 2 A.2 q 2 · 2 , 1 ,c = ,b p q : , b 1 , a 1 . We consider all four-momenta — including loop momenta 1 ˙ ( α x ,c ,b · of α ’s over Dirac’s , b , 2 · 2 p 2.3 , σ ) on the left. ˙ ··· ,a α 2 1 , a  7→ ,b 1 a α, 1 A.2  a µ ( ,b p 2 + ). Thus, when a momentum ,a ˙ α 1 , as well as objects such as [ α a
p  b . This notation is somewhat more cumbersome, but unquestionably more q e λ , ]. When discussing the momentum a e λ , we will occasionally refer to the spinor variables for : the above identity implies that = det( 127 a µ p ]. In terms of these, we can define the SL(2) invariants µ a | p : One final, potential simplification worth mentioning is that whenever a bracket involves Notice that the norm squared of any momentum is equal to the determinant = det : = ] = : a 2 Nota bene ab e consecutively repeated indices, they factor out: ( should be understood as being taken over the 2 The generalized square-angle bracket ontion the right to hand most side amplitudes above researchers;in should but be terms for familiar of our nota- the purposes bracket we ( may consider it to be defined [ less) momenta familiar than the new notationmay in arise ( is formassless brackets then involving at least one massless particle. For example, if p full rank and can thereforevariables be [ expressed as an outerby product index λ This definition is reminiscentas of the a ‘Dirac trace’ tracebecause tr for it good makes reason: clearerwe that it happen it can to is prefer alternatively not Pauli’s be cyclically written symmetric, and because in this instance Civita symbol. In terms of these, the trace-like ‘bracket’ is defined to be where Pauli matrices Notice that the(complexified) indices Lorentz group, respectively, and can be raised and lowered using the Levi- A.1 Representing (functionsThe of) basis four-momenta of integrandsbracket we introduced in construct section involve ‘chiral’— numerators to built be from expressed the (without trace-like any loss of generality) as 2 A Explicit numerators for six-point integrands JHEP12(2019)073 9) 5) 6) 7) 8) 1) 2) 3) 4) ...... I I I I I I I I I ( ( ( ( ( ( ( ( (   5 4 , , f f , , g g , , a a , , d d , , 3 3     2 2 2 2 b b g g 1 2 2 2 + d d  + 5  − − ,  4 f 5 2 2 , , , e e g f f 2 2 , , , b a a g g , , , c b , a , − − , c 3 , d 2 2  , 3 d d 4 2 2 2  a  a a 2 2 1 2 b a + +   + − − d d    , , – 24 – a a a h h , , ,  , ,    g g g 4 f f 6 6 , , , 6 , , , , , , e f f f a a h h g , , , , , , ,   f 5 5 4 f f f , , , ,   g      , 5 5 4 3 h 4 3 4 3 , , , , , ,    c c c c c c , , , , , , 3 4 3 , , , b b b b b b , , , , , , c c c , , , 1 1 1 1 1 1 b b b , , ,       1 1 1 − − − − − −    numerator integrand : : : : : : : : : 9 7 8 6 5 4 3 2 1 I I I I I I I I I A.2 Explicit expressions for the numerators of basis integrands JHEP12(2019)073 14) 15) 16) 17) 18) 10) 11) 12) 13) ...... I I I I I I I I I ( ( ( ( ( ( ( ( (   5 , e , d , c , b , 1  ) 2 h + 16 2 s g 2 e 2 +(  d   1 2 h h , ,  g g − , , 6  , 1 1 , , 5 a b b , , , , d f c c   , , , , g 6 4 3 d d , , , , ,  a a e e f 2 , , , , , b d d g 5 4 , , ,   2 4 − 1     − − 2 2 – 25 – 6 b 1 2 d   ,  h h 3 , , + + − , g g e    , , , a a 6 5 4 f , , , , , , g g e e 5 (4+5)   g , , , , , , , 6 6 f f d d 4 3 h , , , , , , , , , g g c c 4 5 c c c , , , , , , ,   b b f f b b b , , , , , , , 1 1 12 12 4 3 1 1 1   s s        12 12 − − − − − 1 2 1 2 s s : : : : : : : : : 18 17 16 15 14 13 11 12 10 I I I I I I I I I JHEP12(2019)073 23) 24) 25) 26) 27) 19) 20) 21) 22) ...... I I I I I I I I I ( ( ( ( ( ( ( ( (    5 2 2 , g g a 2 2 , d d d , 2 − −  2 2 2 e e h  2 2 − a a 3 ,  f − − 6 , , g 2 2 a , , d d 1 d 2 2 ,  a a 2 2 – 26 – d + +  2   + e d d  , ,   a − h h 3 4 , , , , ,  g  e e f f , 6 56 456 , , , , 6 , f s s , a a f f , h , , h   ,   3 c c , , , f 4 3    f , , , b b , , , c c 5 , , 12 12 12 4 1 1 b b  s s s  , ,   1 1 12 12 − − − − −   s s : : : : : : : : : 27 26 25 24 23 22 20 21 19 I I I I I I I I I JHEP12(2019)073 33) 34) 34) 36) 37) 28) 29) 30) 31) 32) ...... I I I I I I I I I I ( ( ( ( ( ( ( ( ( (  13 s 2 g 1 2 16 s  − 2 2 , h  e 2 h , , d f g , , 1 − 3 ,  2 1 13 16 h   s s 2 2 2 + , 1 2 g h e  , 2 2 − d d d f , ,  e 1 2 , − a 1 , 6   b  , 2 − , h 4 1 h , 3 ,  ,  g e 2 , , 2 2 , + 6 d e  f , ,  , 1 2 1 2 2 – 27 – f  , ,   2 e g 2 2 , , f + a d f 26  , + s + + g 2 , , 2   e a  , h h h , , 1 2 f , , g g g , , g , , + 6 4 5 , , (5+6) (4+5+6) 6 , , , , e e e , , 45 16 1 1 1 , d d s s d , ,    , 1 1 2 12 2 14 12 12 1 − − −   s s s s  : : : : : : : : : : 37 36 34 35 33 32 31 30 28 29 I I I I I I I I I I JHEP12(2019)073 (B.2) (B.3) (B.1) external n . : β lists of external legs ) of the (unordered) = α -1 δ a , = ]. From the : , β {} ... ) is absolutely ordered, and ordered , α 128 1 f β , ~a are chosen; in terms of these, a . ( ~a : , β α = ( MHV (three-point) amplitudes. ~a and , where YM n ) will always be taken to mean MHV × A α : {} = B.3 ~a to represent the color-dressed amplitude. = ) becomes graphically, α β β f ~a ) -1 a A B.1 YM n ( ) -1 S , will be denoted by ‘flat’ vertices according to A e A X ∈ ( – 28 – f ~a S YM n X ∈ ~a is not cyclically symmetric. Finally, notice that we A ··· 2)! permutations ) = 2 = ), e ~a β − , α β 2 a f n ). Notice that (following notational conventions familiar A 1 B.1 , e A , two arbitrary legs α f ( ( ) denotes 1 e S n YM 1 YM n a B.1 A ) to include the case α A sets of legs are indicated in black. Also following familiar conven- f B.2 i e X users) we have used ‘-1’ to denote the final entry of an ordered list. The . , denoted = } : ) should be understood with some care: are structure constants in some Lie algebra. Notice that the graphical β ~a , B.2 unordered α β B.2 α f a b c f \{ ] consists of a single leg) for six-point MHV amplitudes in sYM and SUGRA. We will start with Yang-Mills n ~a 2 , while = [ : As already seen above, we have used ‘spherical’ vertices to denote fully Bose-symmetric A Mathematica blue The careful reader will noticein that we are consistently coloring tions, the blue vertices indicatingamplitudes, amplitudes while in white ( vertices will be taken to mean pearing in the DDMthe expansion usual ( conventions. Thus, the expansion ( notation of ( (unless may liberally extend ( tree amplitudes in both sYM and SUGRA. The color-ordered partial tree-amplitudes ap- color-factor appearing in ( where the where the sum is takenset over the ( to We may disentangle thedescribed color by structure Del fromparticles Duca, the of Dixon kinematics an and amplitude (non-uniquely)we Maltoni in can (DDM) the represent the in way amplitude ref. as [ B.1 Color-dressed leadingLet singularities us of start MHV with amplitudesmay a in be represented discussion sYM and of sewn together howin into Bose-symmetric gauge unambiguous (tree) color-dressed theory. leading amplitudes Let singularities involving us gluons use a calligraphic In this appendix, we give explicitin representations table of all the leadingtheory singularities enumerated because the presencemachinery of to non-kinematic be color-factorsenumerated developed. requires in a bit Formulae more for technical SUGRA are comparatively simple, and are B Explicit leading singularities for sYM and SUGRA JHEP12(2019)073 . ]. ~c 77 (B.6) (B.7) (B.8) (B.9) (B.4) (B.5) and . b 4 ~ 4 e e 6 5 e e . 2 2 e e f . f 6 6 e  b e ~ b ~ 5 R 5 e e ~c f , f , 4 4 e R ,  ~c e ~c ), but with a bit more 3 b ~  R 3 e , ) e f ~c 1 f expansion, if desired [ R , 2 b B.6 2 ~ e | c ~a R e ~a ~a δ 1  b | 1 ~ e f e , 2 /N in ( f f ~c R 5 ( 6 × = e  ) it is extremely straightforward e , ~a 3 c ~c )  3  e , e 1 e ~c b N f 1 ~ , ~c 1 e , | e R f ~a γ = f |  ~a ), it is easy to see that the color-factor 2 i  e , i f b e ~ e ( ~a R X X , , b B.7 ~ b ~ ) symmetry generated by the equivalences =  , = β f ~c | | 2  b . ~ ~a | Z e | f ~a + ) ) ) | α – 29 – × ( A C B ~a = ( | ( (  3 S S S f  X ∈ ∈ ∈ S ~c ) ) ~a , 2 2 = b ~c ~ | | ~a 1 1 , = (-1) b ~c ~ ( ( b ~   e = ~a f , b ~ = = = , ~c    ~c ~c f , , b ) would be expressed R ~ ) is the ‘reversal’ of = , b ~ 1 ~a , ) is an ordered list of external leg labels, and similarly for   B.4 a ~a ) are sufficient to represent all contributions at two loops. We could -1 , ~c e f ,  a b e ~ f , , | ... B.4 b ~ , | ~a  ... -1 , )’ denotes the concatenation of ordered lists. For any particular Lie algebra, f a 1 a = : = ( = (-1) :  : = ( ~c : R  ·| · · · |· , b ~c ~a ~ ~a , , b ~ ~a For our own purposes, we find it more convenient to introduce a (notationally) less- This graphical notation can be used to represent color-dressed leading singularities in ,  ~a e f  f where ‘( the structure constants definingdecomposed however these one may color wish. factorsto For express can the each case be of of these SU( combined in terms and of summed, multi-trace terms and in the 1 graph appearing in ( These color factors enjoynotational the complication same (for symmetries which as we apologize): In terms of the color-factor function defined in ( manifestly-symmetric color-factor function where These ‘symmetric’ color-factors enjoy where in the example ( define a (maximally, manifestly) symmetric color-factor Let’s discuss each of these ingredients inB.1.1 turn. Color factorsIt at is two not loops: hard to diagrammatics see that & color notation structures with the topological structure of those appearing sYM as (sums of) color factors times kinematic factors; for instance: JHEP12(2019)073 ~σ ), ), we B.10 (B.11) (B.12) (B.10) . B.12
a orderings of e η a λ a . P is that it is equal  cyclic ) 4  δ . × ~c | 2 ,  2 δ . ~c ~c
, | i a b ... ~ p 1 α , , | An explicit definition of Γ a σ ~a 1 ~a   ~c -1 | P 4 σ γ 2 ( × , , 2 1 ) δ ) γ | i · · · h ~σ ~c 2 2 b ¡ ~ σ | 1 consistent with the β ··· PT( | σ 1 ¡ h ~σ b b invariant. Finally, as already emphasized ~ ~ involving a cyclically-ordered set of legs | ¡ δ 3 = 5 : ( ~a X ) , S ∈ ) -1 ~σ β | = σ cyclically invariant. – 30 – , ~a  | ~c α , ... ( ,  1 Γ ... σ : , ) above: ~a =  , Γ’s arguments include repetition. For example, in ( Γ separately B.4  MHV vertices). For two-loop MHV leading singularities, Γ . More formally, we could write ~c , ~c , b ~ ) = PT( , ~σ ~a ...  , f MHV vertex. Also notice that the ordering of the arguments of Γ ~a for all cyclic permutations PT( ’ is being used (somewhat unusually) to denote what was described
], to which we refer the reader for a more thorough discussion; however, ¡ ~σ appears in its first and third entries, as both of these MHV vertices involve 59 α ] for color-ordered MHV amplitudes through the 32 α Notice that, unlike Being built from ordered MHV amplitudes, theyWe also are enjoy neglecting a the dihedral factor for symmetry, but supermomentum conservation at the possible ’ is admittedly a bit unfamiliar, we believe it to be in the same spirit of ordinary shuffle 4 5 ¡ ‘ sums, and therefore have allowed ourselves this slight abuse of notation. Usingcost of ( an overall sign. where the symbol ‘ above: namely, all ‘cyclic shuffles’ ofof each ‘shuffle’, list. duplicated Notice elements that, ofterms unlike distinct other and lists uses may are and appear; definitions not they repeated must in be aligned the between shuffle. Although this usage of shuffle and the symbol to the sum ofeach PT of the arguments formula [ The first and perhaps most conceptually simple formula for Γ the leg has absolutely no meaning, andabove, the so Γ arguments is of fully was Γ given are in ref. [ it is worth quoting the two primary formulae derived there. Recall the famous Parke-Taylor external legs involved in eachconnected MHV to (blue) vertex it of via thewill a graph always chain (including depend of any on external exactly legs three (cyclically) ordered liststhe Γ leg labelled We would like to define a function Γ which depends on the (clockwise) cyclic ordering of Let us now discuss thethe summand kinematic of parts the of example leading ( singularities such as those appearing in B.1.2 Kinematics: on-shell functions of ordered (MHV) amplitudes JHEP12(2019)073 ) (PB) (KB) 1 ! ) ), and (HBa) (B.13) ~c β , γ , ]), a more ↔ HBb 2 b ~ 91 α , ! β +( ,  a b )  , ) δ ) | 1 γ α 2 ! b | ~ | ) and B ( ~c 3  δ | | ) ~c | 2 × α γ | ~c PT( | β  ( 1 | ) 2 HBa ¡ ~c × 2 1 ~c | | ~a ~c  b ~ | γ | ∈ 2 X β ( | ) ; (B.14) δ a b α ~c | , | 1 ~c , × ) 2 1 ) ~c =  | ~c ~c γ 1 | ( 1 | ~c α δ 2 , | ~c ( ( ) b , ~ δ )PT( , | , 1 | ) ) b ~ ) ): b b ~c β ~ ~ | | | | δ α | | 1 γ γ γ b b | | | b ~ ~ ~ | | | 3 2 2 DPb δ γ γ )PT( b ~c ~c ~ ( ( ( ( ( ( ~a , , , , , , ), ‘hexaboxes’ A ( ) ) ) ) ) ) β β β β β β | | | | | | PT( PB ~a ~a ~a ~a ~a ~a | | | | | | × – 31 – α α α α α α ( ( ( ( ( ( 2 = ) and B (       J  is somewhat involved, and so we refer the reader to Γ f Γ f Γ f ) =

δ J | ) ) ) ) ) ) ) ) ) DPa  2 A A A C C C B B B ) for Γ makes it clear that Γ only ever has single poles in ( ( ( ( ( ( ( ( ( ~c ~c | , S S S S S S S S S X X X ) is used to compute the leading singularities of sYM in α ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ | ) ) ) ) b b ~a ~ ~a ~a ~ 2 2 3 2 1 ... B.12 b ~c ~c ~c , ~ | | | | ~c | 1 1 2 1 ~a b B.14 ), ‘pentaboxes’ ( ~c ~c ~c ~ |  ( ( ( γ 1 ( Γ ~c ( , = = = ) KB γ | 2 b ~ | ) as β | ’ denotes the concatenation of ordered lists. 1 | b ~ | B.10 δ ( , ) β | ~a | ] for details. It is worth remarking that, for the sake of computational efficiency, α ( 59 While the representation (  Γ There are onlymultiplicity. six classes It of isthem leading not as singularities hard ‘kissing boxes’ for to ( ‘(non-planar) give two-loop double a pentagons’ MHV A closed amplitudes ( formula at for all each. We colloquially refer to only the compact formulathe ( supplementary material for this work. B.1.3 Closed formulae for all two-loop MHV leading singularities compact (and algorithmically more efficient)is formula simply for Γ also exists. This representation the definition of the Jacobianref. factor [ where as before ‘ the external kinematics (and is therefore ‘local’ in the sense discussed in ref. [ could express ( JHEP12(2019)073 . ). A (DPa) (B.15) (B.16) (B.17) (DPb) (HBb) B.12 . (  ) ) ~c α | b ~ | HBb β ( ,..., , ) ~a β | ! .  ~a . | ) α ! β ( |  An important and useful  2 ! ! ) ) is entirely well behaved ~c   | in terms of sums of Parke- β ) ) , = Γ × α  | β β α  | | ~c 1 HBb ( ) ~c ~c ~c | | , | β ) | α α × γ 2 ( ( α (  | ~c , , ,..., , ( b ) ) ~ × ) ~a | , {}   α α γ ) , β × | | | ~c 1 ( b b b b ~  , ~ ~ ~ | | | , ~c , ) | ~c ) ) , β β β b ~ γ | ( ( ( β b β | dashed leg ranges are allowed to be empty ~ | | , , , , β b ~ ) ) ) ) ~a ( ( ~a | | ' , , β β β β | | | α | ) ) α ( ( ~a ~a ~a ~a ~a ~a | | | | | | →{}   7−→ – 32 – f Γ α α α α α α C ( ( ( ( ( (

      ) ) f Γ f Γ f Γ A B ( (

S S ) ) ) ) ) ) ) ) ) X ∈ ∈ A A A C C C B B B ( ( ( ( ( ( ( ( ( b ~a ~ S S S S S S S S S X X X = ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ) b b b ~c ~c ~ ~a ~ ~a ~a ~ 2 ~c | 1 ~c ( = = = =  ) β , α ( , ) α | b ~ | β ( , ) β | ~a | α (  Γ If this is not entirelyTaylor obvious, factors recall consistent the with definition the of cyclic Γ ordering of each ordered vector It is very easy to see that the kinematic factor is equivalent to a one-loop leading singularity: in this limit, and in fact reduces to a simpler case: A few moments thought will demonstrate that the formula ( feature of the formulae enumerated aboveWhat is does that an ‘empty’ legto range understand imply than physically may or be functionally? expected. It Consider turns out for to example be ‘hexabox easier B’ ( Smooth degeneration to composite leading singularities. JHEP12(2019)073 ] , 2 130 , (LS1) (B.18) 129 ). As the 6 i what would be 56 MHV) amplitudes ih [45][56] 45
exactly i ih 6 | 36 1 | ih 3 | 2 34 | ih 26 ), the only difference between this ih (5+6) | 24 4 ih B.16 23 i − h 6 ih | 2 16 | = : 3 ih | – 33 – 1 | 14 ih one-loop leading singularity is the color factor. This 13 (5+6) ih | 4 h 12 h actual = ], the actual expressions found for leading singularities in SUGRA ) and an 95 B.16 ], and the one-loop leading singularities, 130 The only ingredients needed for us are the six-point MHV tree amplitude in Indeed, on some level, the only known way to compute the leading singularities of Due to this aspect of the kinematic part of ( In the expressions given below, the signs of the were determined via consistency with sYM. 6 at the vertices, we canto make obtain use analytic of expressions. the tree-level formulae derived by HodgesSUGRA in [ [ SUGRA is the generalthe way: solutions to evaluate all thedefinitional the cut approach is equations, tree guaranteed and amplitudes to alwaysleading sum involved work singularities at (up over needed to for all the perhaps MHV states vertices amplitudes an involve that overall on only sign MHV can (and be exchanged. This There does not yet existshell any functions truly satisfactory in approach SUGRA. todimensions) Although the there computation (as exists of with suggestive general representations allmannian on- of massless integrals on-shell quantum functions [ field instill terms theories surprise of in Grass- us four with their simplicity. we drew eachbivalent figure (‘empty’) MHV in vertex, a andenumerated apply way above. the to appropriate formula emphasize from this the general fact: cases B.2 replace any dashed Two-loop leading line singularities with of a MHV amplitudes in SUGRA where the momentum flowingleading through singularities the dashed needed line insimply vanishes. the special Thus, representation cases all of of the six the composite particle six amplitudes general topologies in enumerated sYM above. are Indeed, in table expected if one approached the composite singularity the summand. degenerate case ( is a good thing,one-loop as amplitudes. two-loop Indeed, amplitudes have the different color powers factor of is the easily coupling seen relative to to be two element list is cyclically consistent with all permutations, and thus has no affect on JHEP12(2019)073 ) ) ) ) ) ) ) ) 8 4 8 5 8 6 8 7 8 8 8 1 8 2 8 3 f f f f f f f f ( ( ( ( ( ( ( ( (LS2) . i 56 ih 46 i ih i 46 i 45 i i ih i i 46 46 ih 56 56 ih 45 ih 45 56 [12][23] [12][45] i i 35 ih ih ih ih i ih [23] [23] 45 3 45 3 = 8 supergravity: 2 2 | | ih [23] 45 46 ih 36 ih 36 56 45 2 N 34 ih ih ih ih ih ih 34 34 [56] [56] ih i i 35 45 ih 35 ih 35 45 34 [56] 1 1 (1+2) (5+4) | | i | | 26 [12][23][45] [12][23][56] ih ih ih ih ih ih 1 24 24 i i 6 | [6 | ih 3 3 i 34 34 ih 34 ih | 34 36 24 | [12][23][56] 3 . With these, we find the following expressions 23 | i ih ih ih ih ih ih 23 23
(2+3) (2+3) 1 ih | | e – 34 – η | · 23 23 ih 23 ih 23 34 23 (2+3) (4+5) λ [4 [4 | 16 | ih ih ih (4+5) ih ih ih (4+5) i i 16 16 | | [4 8 ih 4 4 [3 (1+2) | | i in the case of 16 16 ih | 16 ih 16 23 16 [6 [6 × 4 i i (2+3) 2 13 | 2 ih [5 ih ih ih ih ih | [12][23][45][56] 1 1 15 15 δ i | | ih
[4 1 ih 15 ih 15 14 14 15 16 | e λ 456 · 12 (2+3) (2+3) ih ih ih ih ih ih s | | h λ 13 13 456 (2+3) s [1 [1 | ih 14 ih 13 13 13 13 14 2 (2+3) (2+3) | | × [1 ih ih ih ih ih ih (2+3) 12 12 2 = | [6 [4 h h δ 12 12 12 12 12 12 [4 h h h h h h − − ======5 5 5 4 5 5 4 6 6 5 5 6 4 4 6 6 6 6 1 4 1 3 3 1 4 1 1 3 1 3 1 4 2 3 2 2 3 2 3 2 2 2 ======8 8 8 7 5 8 6 8 4 8 8 3 8 2 8 1 f f f f f f f f momentum conservation, for the leading singularities of table As for sYM, we have systematically neglected the ubiquitous factor imposing super- and JHEP12(2019)073 ) ) ) ) ) ) ) ) ) ) 8 9 f 8 14 8 15 8 16 8 17 8 18 8 10 8 11 8 12 8 13 f f f f f f f f f ( ( ( ( ( ( ( ( ( (
i [36] i 56 34 i ih ih i i 56 46 16 ih ih 46 56 h ih ih 46 [23][45] 45 i ih ih 45 36 6 [26]- | i ih ih 45 36 46 ih ih 34 35 ih [23][45][56] [12][34][56] ih ih 36 34 i i (4+5) i | 1 3 12 ih 3 ih | 24 | 34 h | [6 i ih ih 26 26 1 i | i ih ih 23 23 4 | – 35 – 13 i (4+5) ih (1+2) ih 23 16 5 (4+5) h | | | | ih 14 ih 16 16 6 6 [6 [4 h | | i i (2+3) ih ih | 16 1 1 12 12 12 12 4 1 15 h h | | s s s s [4 ih [12][23][34] ih 15 15 i i × × × × × × 6 ih 15 ih 6 | 14 | ih ih 14 13 (5+6) (2+3) | | ih 13 ih 12 [2 [1 h ih (2+3) 12 12 (1+2) | h | h 12 [1 h [5 − − ======5 4 5 4 3 4 4 3 5 5 6 6 5 6 4 4 5 6 5 4 6 3 4 6 5 4 5 3 3 3 6 5 6 6 6 3 1 4 1 2 1 1 2 3 3 1 2 2 1 1 2 1 2 1 1 3 2 2 2 2 ======8 9 f 8 18 8 17 8 16 8 14 8 15 8 11 8 10 8 13 8 12 f f f f f f f f f JHEP12(2019)073 ) ) ) ) ) ) ) ) ) ) 8 24 8 25 8 26 8 27 8 28 8 19 8 20 8 21 8 22 8 23 f f f f f f f f f f ( ( ( ( ( ( ( ( ( ( i 2 | – 36 – i 1 | 12 6 h | 1 12 12 12 123 12 12 12 56 123 h s s s s s s s s s × × × × × × × × × × ======5 3 5 4 4 4 4 5 5 6 2 4 3 5 4 6 4 3 5 3 4 5 3 6 4 5 6 6 3 5 3 1 3 1 4 6 3 6 5 1 2 6 6 6 2 1 2 1 1 1 2 2 1 2 1 3 2 2 2 1 ======8 28 8 27 8 26 8 24 8 25 8 23 8 22 8 21 8 19 8 20 f f f f f f f f f f JHEP12(2019)073 ) ) ) ) ) ) ) 35 37 8 29 8 30 8 31 8 32 8 33 , , f f f f f ( ( ( ( ( 8 34 8 36 f f ’: (B.19) ( ( b 16 s 12 2 14 s s × × . 8 ∞ f : = i = = 46 →∞ b 4 ih 6 i = Res 45 5 5 6 12 ih h 4 3 36 3 2 ih 2 1 i 35 5 1 i i | ih 2 1 = = 8 supergravity, there are three poles at | | = i 16 1 i | 8 35 4 i i ih 8 37 | f N 25 i f (3+4) h 3 | | – 37 – 12 12 15 24 h (2+3) h 1 1 In , h | | | ih (5+6) 1 2 2 12 | | h h [12][23][24][25][26] s 45 14 , 1 2 s h h ih × × × 2 12 12 s s × × 13 →∞ h b × × = Res = − − − ======4 3 4 5 2 3 3 4 4 6 2 5 3 6 4 5 4 3 3 = 4). Surprisingly, the value of the residue at infinity is the same in all 6 6 5 3 6 6 N 6 5 4 2 2 2 5 1 2 1 2 1 1 1 1 1 →∞ ======Res b 8 34 8 36 8 33 8 32 8 31 8 29 8 30 f f f f f f f three cases. The violatedby residue making theorems use can of be Cauchy’s obtained theorem from in the the following remaining hepta-cuts unfixed variable denoted by ‘ Poles at infinityinfinity in that supergravity. are relatedsatisfied to in the failure of the following three residue theorems (that are JHEP12(2019)073 5) , point 1 , 2 , 4 , six 3 , — linked to on (6 8 11 f − 6) (B.20) , arXiv 6) , notebook file illustrat- 4 , 1 , 5 , 2 , 3 , 4 , 2 , 3 The ancillary file , (1 package file, consisting of code (5 8 21 sYM and SUGRA as represented f 8 11 . data file, consisting of (just) the f consists of 5 elements: − − 4) , 5) , both 6) 3 in either sYM or SUGRA. , , Mathematica 4 , i 5 1 conventions of the previous indices and f , , 6 ). , 3 2 , , : a 3 N i , 4 5 f , , 2 , plain text 2 B.20 , (6 (1 Mathematica (1 8 33 – 38 – f 8 6 8 12 f f : a − : a − − 4) , 6) 6) , , 3 , 5 4 a list of 37 permutation-seeds which generate the six- , , 1 , 4 5 : , , 2 , 3 3 , , walkthrough.nb data.dat tools.m 6 , 2 2 chiralIntegrandSeedData , , (5 (1 (1 ). These contributions are encoded somewhat indirectly, and it 8 6 8 33 f f : a symbol representing 8 12 f ] 1.1 − − enumerates all of the ingredients required to use and verify the − 6) 6) , , 6) , 5 5 , , 5 integrand integrand integrand , 4 4 , , 4 , 3 3 , , 1 legList , 2 1 , , , 2 , i data.dat (1 (2 [ (3 point point point ). We have explicitly checked that these are the only residue theorems that 8 20 8 6 8 20 f f f f 1. = = = B.19 in the summand ( is worthwhile to describe how eachEach term element is of represented. the list chiralIntegrandSeedData point two-loop MHV amplitude integrands for ing our results and the main functionality of the codebase. details of our result described in these appendices.six useful to analyze, understand, and work the thesix raw data. six 8 ∞ f • • • • integrand results described in this work. These ingredients consist of the objects: These files are all well-documenteddescribed within, below. but the key data structures inMain the results: data six-point file MHV are amplitude integrands. files can be downloaded fromthe the right-hand abstract panel page (below for ‘Download’).interested this reader: Specifically, work we on have the prepared three files for the supergravity are linked to the topologies in ( C Organization of the supplementaryThe material main results described in this work are available as supplementary material. These Plugging in the explicitities formulae for ( supergravity givenare above, violated one reproduces in the supergravity, so equal- that the only non-vanishing poles at infinity originate in diagrammatic interpretation. We willbut not summarize write the these contributingalso terms terms indicate out in the in the relevant permutation a of graphical external notation, legs: A number of residues will correspond to regular factorization poles that have an on-shell JHEP12(2019)073 , ] } ∈ ] 2 7→ i c , a , v 1 { b , v { a [ to the edges = 8 SUGRA. 2 , ` hep-ph/9403226 N 1 = 4 sYM. These , and etc. gamma ` [ of the form ] are labelled by in j N = 8 supergravity. , and i 8 i f i p ] in N point gauge theory [ c — exactly matching the Rules 4 i : the first three on this , n sb f : b } (1994) 217 , a ] = . They are expressed directly [ ij of the vertices are chosen from ,[ . ] } B 425 One loop a, . . . , h 6 j Nota bene { , colorF i . . They are expressed as sums of prod- [ Heads N i f ab , . . . , p : 1 = Mathematica i Nucl. Phys. edges of for graph, encoded as pairs of . The , ij 2 ), respectively. h . – 39 – v MHV, external, or ‘soft’, respectively. This list of . 7→ ) which are not identities in B.12 A.2 a, . . . , h, p { Mathematica 1 v oriented A.2 B.20 ), which permits any use, distribution and reproduction in 1 depending on whether edge has been oriented as incoming : the numerator for this integrand in the prescriptive basis, ) and ( ∓ for MHV, : one possible assignment of loop momenta } B.7 of labels s , : a list of : for each vertex in the corresponding graph, a list of edges incident p , A.2 a list of permutation-class representatives of residue theorems : w CC-BY 4.0 , ; and internal edges are labelled an explicit formula for the leading singularity b } { 6 This article is distributed under the terms of the Creative Commons . Again, we should emphasize that nothing about our result depends on ]: an explicit formula for the leading singularity i ]: [ ]. ···} ,..., , 1 1 i ` this assignment of internal loop momenta. as enumerated in appendix conventions of appendix for an edge connecting the set edges has duplicated entries inwith the case this of duplication, soft the (composite)in edges leading appendix are singularities; always listed according to the assignments of the graph consistent with momentumbut conservation and chosen) the orientation. graph’s (arbitrary, This is encoded as a list of with it, with a sign or outgoing, respectively.{ Edges for external momenta [ integrandNumerator graphEdgeList loopRoutingRules graphDialList SPIRE 5. 4. 3. 2. IN amplitudes, unitarity and collinear[ limits list correspond to those of ( without relying on any functionalityin of terms of familiar spinor products grtGenerators (‘grt’s) satisfied by the on-shell functions relying on any functionality of ucts of terms builtdefined from in the equations (abstract) ( objects sugraLS These are given separately and explicitly so that they may be used by researchers ymLS are given separately and explicitly so that they may be used by researchers without Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, • • • [1] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. JHEP12(2019)073 , , = 4 , Nucl. JHEP = 4 N , , , ]. N (2012) 081 ]. SPIRE 04 ]. ]. IN , ]. ]. ][ , SPIRE SPIRE JHEP IN SPIRE to four partons IN , SPIRE SPIRE ][ IN − ][ IN IN e ][ ]. + ][ ][ ]. e (2014) 030 ]. (2007) 125020 ]. Maximally supersymmetric SPIRE 10 SPIRE ]. 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