The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory
aa r X i v : . [ h e p - t h ] S e p Saclay/IPhT–T10/096Brown-HET-1607
The Six-Point NMHV amplitude in Maximally SupersymmetricYang-Mills Theory
D. A. Kosower
Institut de Physique Th´eorique, CEA–Saclay,F–91191 Gif-sur-Yvette cedex, France ∗ R. Roiban
Department of Physics, Pennsylvania State University, University Park, PA 16802, USA † C. Vergu
Department of Physics, Brown University,Box 1843, Providence, RI 02912, USA ‡ Abstract
We present an integral representation for the parity-even part of the two-loop six-point pla-nar NMHV amplitude in terms of Feynman integrals which have simple transformation propertiesunder the dual conformal symmetry. We probe the dual conformal properties of the amplitudenumerically, subtracting the known infrared divergences. We find that the subtracted amplitudeis invariant under dual conformal transformations, confirming existing conjectures through two-loop order. We also discuss the all-loop structure of the six-point NMHV amplitude and give aparametrization whose dual conformal invariant building blocks have a simple physical interpreta-tion.
PACS numbers: 11.15.Bt, 11.15.Pg, 11.25.Db, 11.25.Tq, 12.60.Jv ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: Cristian˙[email protected] . INTRODUCTION What is the scattering matrix of a matter-coupled gauge theory? In general, this is ahard question involving both conceptual and technical subtleties. Nevertheless, scatteringamplitudes enjoy a much simpler structure than implied by their expansion in terms ofFeynman diagrams. For some theories, additional off-shell and on-shell symmetries simplifythe amplitudes enormously. The further simplification exhibited in the planar (or infinite-color) limit may even allow a complete answer to the question.The N = 4 or maximally supersymmetric Yang–Mills theory (MSYM) may be such atheory. The simplifications inherent in the larger symmetry have already allowed explicitcalculations of scattering amplitudes well beyond those for other theories. At weak cou-pling, advances in multi-loop and multi-leg calculations [1–7] have opened the possibility ofprobing the structure of the scattering matrix to high order in perturbation theory. TheBDS conjecture [8] for the all-loop resummation of maximally helicity-violating (MHV) am-plitudes (based on an earlier relation [9] linking one- and two-loop amplitudes) provides anexample of possible structures that can emerge. At strong coupling, the leading expansionof scattering amplitudes has been computed using the AdS/CFT correspondence [10] by Al-day, Gaiotto, Maldacena, Sever, and Vieira [11–14]. This two-sided approach, together withthe recent developments in the evaluation of scattering amplitudes at strong coupling forany number of external legs and the realization that the relation between certain scatteringamplitudes and null polygonal Wilson loops carries over from strong coupling [15] to theweak-coupling regime [16–20] offer circumstantial evidence that the N = 4 super-Yang-Millstheory may ultimately be solvable in the planar limit.Gluon MHV amplitudes, with two external legs of negative helicity and the rest of positivehelicity, are the simplest amplitudes in a gauge theory. They are particularly simple in theplanar limit of MSYM, where they are determined by a single helicity structure, accompaniedby a function of scalar and pseudo-scalar momentum invariants, to all orders in perturbationtheory. This simplicity is of course shared by their parity conjugates, the MHV amplitudes.The structure of the remaining non-MHV amplitudes is more complicated. At one loop,explicit expressions are known [21, 22] at arbitrary multiplicity for next-to-MHV (NMHV)gluon amplitudes, with three gluons of negative helicity from the rest. These are alreadymore intricate, with the number of independent helicity structures growing cubicly with the2umber of external legs, each multiplied by an independent function of scalar and pseudo-scalar invariants. How does this structure generalize to higher loops? No explicit expressionsare known to date for higher-loop non-MHV amplitudes. In this paper, we take a first steptowards filling this gap, computing the parity-even part of the two-loop six-gluon NMHVamplitude. These results were first reported in ref. [23]. This amplitude, which comes withthree inequivalent helicity configurations, is the simplest non-MHV amplitude.General results on the structure of infrared divergences in massless gauge theories suggeston one hand, that the divergent terms have a simple iterative structure; and on the other,that all planar amplitudes with fixed number of external legs share the same structureof infrared-divergent terms. Together with the structure of the splitting amplitudes, thisimplies that a natural way to extract infrared-finite quantities from non-MHV amplitudes isto divide them by the MHV amplitudes with the same number of external legs. Drummond,Henn, Korchemsky, and Sokatchev (DHKS) showed to one-loop order that this ratio is notonly finite but also dual conformal invariant for NMHV amplitudes [5, 24] and conjecturedthat the same holds to all orders for all non-MHV amplitudes [24]. Here we clarify andtest this conjecture to two-loop order for the six-point amplitude. This test requires the useof Dixon and Schabinger’s result [25] for the O ( ǫ ) terms in the one-loop NMHV six-pointamplitude.As an intriguing consequence of the semiclassical approach of Alday and Maldacena [11],anticipated by the structure of flat space string theory scattering amplitudes at high energyand fixed angles [26], to leading order in the strong coupling expansion all scattering ampli-tudes are (in a certain sense) insensitive to the flavor and polarization of external legs. Whilequantum corrections are likely to alter this conclusion, this structure is surprising from thestandpoint of the intricate analytic structure of the weak-coupling scattering matrix.The arguments of Alday and Maldacena led to the identification [16, 17] of a surprisingrelation between one-loop MHV amplitudes and the one-loop expectation value of specialnull polygonal Wilson loops. This relation was shown to hold at two loops as well forfour-, five- [18] and six-particle scattering amplitudes [19, 27]. Integral representations ofhigher-point two-loop MHV amplitudes are also known [28]; comparison with Wilson-loopexpectation values [29] is hindered, however, by the complexity of evaluating the requiredhigher-point two-loop Feynman integrals. Dual conformal symmetry [17, 18, 24, 30, 31]plays an important role in the relation between MHV scattering amplitudes and Wilson3oops. This symmetry is manifest for the integrands of both MHV scattering amplitudesand Wilson loops, but it is broken by the dimensional regulator. Dual-conformal invariantscan be constructed by using the general structure of divergent terms. A particular patternof spontaneous breaking of the gauge group provides an alternative regularization in whichthis symmetry is restored through natural transformations of the regulator [32].It would be interesting to understand whether non-MHV amplitudes also exhibit a similarpresentation in terms of Wilson loops. A necessary condition is that they exhibit dualconformal invariance upon extraction of infrared divergences. It is possible to argue that, toall orders in the loop expansion, four-dimensional cuts of any planar scattering amplitudein N = 4 SYM, in particular non-MHV amplitudes, have this symmetry. Hints in thisdirection also come from the Grassmannian interpretation of leading singularities; in thatframework it was shown [33, 34] that leading singularities are dual conformally invariant.Whether this symmetry survives in the complete amplitude, in the presence of the termsnot constructible from four-dimensional cuts, is an open question. Here we will see that theparity-even part of the six-point NMHV amplitudes can be expressed in terms of pseudo-conformal integrals, i.e. dimensionally regulated integrals that are invariant under dualconformal transformations when continued off-shell.While the structure of collinear limits of non-MHV amplitudes is somewhat more intricatethan those of MHV amplitudes, the former are governed by the same splitting amplitudesas the latter. The iteration relation for MHV amplitudes [9] suggests that one can captureboth the infrared-divergent parts of non-MHV amplitudes, as well as the amplitudes’ be-havior under collinear limits, via an exponentiation ansatz for all the scalar functions thatcharacterize them. This is similar in spirit to the BDS [8] exponentiation ansatz for MHVamplitudes. Such an ansatz is not expected to hold to all orders. Departures from it arecharacterized by dual conformal invariant functions which have properties analogous to theMHV remainder function [19, 27].We perform the calculation using the generalized unitarity-based method, employing avariety of four-dimensional and D -dimensional cuts to express the amplitude in terms ofsix-point two-loop Feynman integrals. The four-dimensional cuts are evaluated in on-shellsuperspace [35]. This approach automatically takes into account supersymmetry relationsbetween different components of cuts and also offers guidance in organizing the calculation.We find that the (appropriately defined) parity-even part of the six-point amplitude may4e expressed as a sum of pseudo-conformal integrals [30], in close analogy with the four-point amplitude through five loops [8, 36–39] and the parity-even part of the five-pointamplitude through two loops [40–43]. There are some additional integrals in the one- andtwo-loop six-point amplitudes, whose pseudo-conformal nature is less clear. Their integrandsvanish as D →
4, yet their integrals can be nonvanishing in this limit. We evaluate theintegrals using the
AMBRE [44] and MB [45] packages and compute the amplitude numericallyat several kinematic points, related in pairs by dual conformal transformations. The infraredsingularities of our expression have the structure expected from general considerations [46,47]. We have tested numerically the dual-conformal properties of the various finite functionsthat can be constructed from the six-point NMHV amplitude.The paper is organized as follows. We review the tree-level and one-loop six-point ampli-tudes in section II, along with their superspace presentation and their conjectured properties.Most importantly, we identify a canonical separation of the six-point NMHV amplitude intoparity-even and parity-odd components. We expect this separation to extend to all ordersin perturbation theory. In section III, we discuss the expected structure of the six-pointNMHV amplitude to all loop orders, based on our calculation using generalized unitarity.We introduce certain finite functions that characterize the amplitude and are expected tobe invariant under dual conformal transformations. In section IV, we describe some of thedetails of our calculation. We use a superspace version of the generalized unitarity method.We discuss some of the subtle points, and give details on the calculation of two importantcuts. In section V, we present an integral representation of the even part of the two-loopsix-point NMHV amplitude. For completeness we also list the even part of the two-loopsix-point MHV amplitude in our notation. We proceed in section VI to analyze our analyticand numerical results for the amplitude, and to test the dual conformal-symmetry propertiesof the various functions that have been conjectured to be invariant under dual conformaltransformations. We give our conclusions and a selection of open problems in section VII. Note added:
As the writing of this paper was being completed we received ref. [92] inwhich an alternative presentation of the six-point NMHV amplitude was proposed as aconsequence of a generalization of the Grassmannian duality for leading singularities to thefull amplitude. The result also contains a proposal for the parity-odd part of the amplitude.Unlike our result, it is expressed in terms of a basis of chiral, tensor integrals written inmomentum-twistor space. 5
I. REVIEW
The n -point L -loop planar (leading-color) contributions to scattering amplitudes of an SU ( N c ) gauge theory with fields in the adjoint representation may be written as A ( L ) n = a L X ρ ∈ S n / Z n Tr[ T a ρ (1) . . . T a ρ ( n ) ] A ( L ) n ( k ρ (1) , ε ρ (1) ; . . . ; k ρ ( n ) , ε ρ ( n ) ) , (2.1)where we follow the normalization conventions of ref. [19] (which differ from those used inrefs. [2, 24]). The loop expansion parameter a is, a = (4 πe − γ ) − ǫ λ π = (4 πe − γ ) − ǫ g N c π . (2.2)Here λ is the ’t Hooft coupling constant and γ is the Euler constant, γ = − Γ ′ (1). Thesum runs over all the noncyclic permutations of the external legs, each of which carriesmomentum k i and a polarization vector ε i .Choosing a specific helicity and flavor configuration for the external legs reduces A ( L ) n ( k ρ (1) , ε ρ (1) ; . . . ; k ρ ( n ) , ε ρ ( n ) ) to a color-ordered partial amplitude. Every partial ampli-tude can be decomposed into a sum of terms, each of which is a product of a functionensuring the correct transformation properties of the amplitude under Lorentz transforma-tions (henceforth called “spin factor”) and a (pseudo-)scalar function which may be writtenas a sum of L -loop Feynman integrals (the “loop factor”). The spin factor is a rationalfunction of the momentum spinors λ i and ˜ λ i associated to the external legs; the parity-evenparts of the loop factor are functions of external Lorentz invariants alone, while the parity-odd parts also depend on Levi-Civita contractions of the external momenta. One could ofcourse choose to re-express the Levi-Civita contractions in terms of spinor variables.MHV amplitudes have two negative-helicity, and any number of positive-helicity, externallegs. These amplitudes in MSYM have the simplest structure of all amplitudes: they havea single spin factor, which is equal to the tree-level scattering amplitude. Computing the L -loop MHV amplitude thus amounts to finding the ratio M ( L ) n ≡ A ( L ) , MHV n A (0) , MHV n . (2.3) We normalize the classical action so that the only coupling constant dependence is an overall factor of g − Y M . This definition of the loop expansion parameter extracts the complete dependence on the Euler constantfrom the momentum integrals. M n in terms of spinor variables, see refs. [48, 49].)All-gluon NMHV amplitudes have three external legs of negative helicity, and any numberof positive helicity. They are the next-simplest amplitudes after the MHV ones. The five-point NMHV amplitudes are MHV; the simplest distinct ones appear for six external legs.These have three independent helicity configurations. In contrast to the MHV amplitudes,NMHV amplitudes contain several distinct spin factors; their forms depend on the helicityconfiguration of the external legs. As a consequence of relations between spin factors, thereare many possible presentations of the tree-level amplitudes. We can single out a canonicalform by constructing the corresponding one-loop amplitude and taking the form that appearsas the coefficient of the double pole in the dimensional-regularization parameter ǫ . Thisrelation [50–53] is guaranteed by the general theorems governing the factorization of softand collinear divergences. We will focus here on the the six-point amplitude. A. The Six-Point Gluon Scattering Amplitude at One Loop
All six-gluon NMHV amplitudes may be obtained by applying cyclic permutations andreflections to the three independent helicity configurations (+++ −−− ), (++ − + −− ) and(+ − + − + − ). The one-loop amplitudes for these configurations were first obtained in ref. [2]through O ( ǫ ) (see also ref. [22]). They can be expressed in terms of three different spinfactors. The spin factors for the ‘split-helicity’ configuration (+++ −−− ) are, B = i s h i h i h h , (2.4) B = i h h i h i h s + i h i [23] h i h h s , (2.5) B = i h h i h i h s + i h i [12] h i h h s ; (2.6)we refer the reader to the original paper [2] for the spin factors of the other independenthelicity configurations. In all cases, the spin factors are uniquely determined by cuts inthree-particle invariants.The six-point one-loop NMHV amplitude for the (+++ −−− ) helicity configuration is7 IG. 1: The integrals contributing to the six-point one-loop MHV and NMHV amplitudes. Anarrow marks the leg with momentum k ; the remaining momenta follow clockwise. The one-massbox I and two-mass easy I integrals contribute to the MHV amplitude and the one-mass box I and two-mass hard I integrals contribute to the NMHV amplitude. The one-mass pentagon I and the hexagon I hex have numerator factors of µ (the square of the ( − ǫ )-dimensionalcomponents of the loop momentum), and hence are finite. They contribute to both the MHVand NMHV amplitudes only at O ( ǫ ) and higher ( I hex contributes to the even parts while I contributes to the odd parts). given by, A (1) , NMHV6 (+++ −−− ) = 12 (cid:16) B W (1)1 + B W (1)2 + B W (1)3 (cid:17) + O ( ǫ ) , (2.7)where, W (1)1 = − X σ ∈S (cid:18) s s I ( σ ) + 12 s s I ( σ ) (cid:19) + O ( ǫ ) ; (2.8)and the sum runs over the permutations, S = { (123456) , (321654) , (456123) , (654321) } . (2.9)All permutations in S leave the spin factor B invariant. The integrals in eq. (2.8) areshown in fig. 1. The factors of in the summand in eq. (2.8) are symmetry factors neededto compensate for double counting in the summation over S . The expressions (2.7) and(2.8) hold only through order O ( ǫ ). At O ( ǫ ) eq. (2.8) receives contributions from additionalintegrals while equation (2.7) receives contributions from additional spin factors. The termsof higher order in ǫ have been computed only recently [25].The other two scalar functions, W (1)2 and W (1)3 , may be obtained from eq. (2.8) by re-placing the set of permutations S by the sets S and S , respectively, where S = { (234561) , (432165) , (561234) , (165432) } , (2.10) S = { (345612) , (543216) , (612345) , (216543) } . (2.11)8he elements of each of the permutations sets S , S and S leave invariant the spin factors B , B and B , respectively. The union of these three permutations sets, S = S ∪ S ∪ S , isthe set of all cyclic permutations and their reflections; the MHV amplitude can be expressedas a sum over this larger set.The one-loop scattering amplitudes for the other two independent helicity configurationshave a structure similar to eq. (2.7); the scalar functions W (1) i are unchanged while the spinfactors B , B and B are replaced [2] by new spin factors D , D and D for the (++ − + −− )helicity configuration, and by G , G and G for the (+ − + − + − ) configuration : A (1) , NMHV6 (++ − + −− ) = 12 (cid:16) D W (1)1 + D W (1)2 + D W (1)3 (cid:17) , (2.12) A (1) , NMHV6 (+ − + − + − ) = 12 (cid:16) G W (1)1 + G W (1)2 + G W (1)3 (cid:17) . (2.13)Infrared consistency then implies that the tree-level amplitudes for the corresponding helicityconfigurations are [2], A (0) , NMHV6 (+++ −−− ) = 12 ( B + B + B ) , (2.14) A (0) , NMHV6 (++ − + −− ) = 12 ( D + D + D ) , (2.15) A (0) , NMHV6 (+ − + − + − ) = 12 ( G + G + G ) . (2.16)The classic expression for these amplitudes was derived in ref. [55]. In later sections wewill see that the structure present in eqs. (2.7), (2.12) and (2.13) — in which only the spinfactors change between various helicity configurations of the external lines — persists athigher loops as well. B. Superspace and Superamplitudes
On-shell superspace provides a very convenient way of organizing amplitudes in N =4 SYM theory and making manifest supersymmetry relations between them. The bosonicpart of this superspace is parametrized by the usual bosonic spinor variables λ i , ˜ λ i , related to The notable difference between B i and the non-split helicity spin factors D i , G i is that, while the formerare rational functions of products of adjacent spinors, the latter also contain products of non-adjacentspinors. This obscures their transformation properties under the dual conformal symmetry [54], whichbecome manifest only when the amplitudes are combined into a superamplitude [24]. k i by k iα ˙ α = λ iα ˜ λ i ˙ α . The fermionic part is parametrized by Grassmanncoordinates η Ai , where A = 1 , · · · , R -symmetry index. The on-shell fields of the N = 4theory are assembled into a superfield,Φ( η ) = g − + η A ψ A + 12! η A η B φ AB + 13! η A η B η C ǫ ABCD ψ D + 14! η A η B η C η D ǫ ABCD g + . (2.17)A superamplitude is a generating function for the scattering amplitudes of componentfields, which may be identified as the coefficients of the appropriate combinations of η i variables.The component amplitudes may be extracted by multiplying the superamplitude withthe appropriate superfield and integrating over all Grassmann parameters: A n ( k , h ; . . . ; k n , h n ) = Z n Y i =1 d η i Y Φ h i ( η ) A n ( k , η , . . . , k n , η n ) . (2.18)The superfields Φ h i ( η ) have a single nonvanishing term corresponding to the field withhelicity h i .As an example, the n -point NMHV gluon scattering amplitudes appear inside the super-amplitude as follows: A n ( k , η , . . . , k n , η n ) = · · · + η η η A n ( − , − , − , + , + , . . . , +) (2.19)+ η η η A n ( − , − , + , − , + , . . . , +) + · · · , where η is the SU (4)-invariant expression ǫ ABCD η A η B η C η D . In extracting these compo-nent amplitudes, the η variables corresponding to the positive-helicity gluons are suppliedby the superfields (2.17) while those for the negative-helicity ones appear explicitly in thesuperamplitude. Because the half of the supersymmetries manifest in this on-shell super-space can be preserved at all stages of scattering amplitude calculations, eq. (2.19) holds toall orders in perturbation theory.The dual superspace in which the superfield is given by, e Φ(˜ η ) = g + + ˜ η A ψ A + 12! ˜ η A ˜ η B φ AB + 13! ˜ η A ˜ η B ˜ η C ǫ ABCD ψ D + 14! ˜ η A ˜ η B ˜ η C ˜ η D ǫ ABCD g − , (2.20)has also been used, see for example refs. [5, 6, 24, 56]. While the expression for the superam-plitude is unchanged, component amplitudes are extracted by differentiating with respectto selected superspace coordinates, eight for MHV amplitudes, twelve for NMHV ones, etc.: A n ( k , h ; . . . ; k n , h n ) = Y e Φ h i (cid:18) ∂∂η (cid:19) A n ( k , η , . . . , k n , η n ) . (2.21)10or pure-gluon amplitudes, the differentiation is solely with respect to the Grassmann coor-dinates of the negative-helicity gluons. The structure of superfields is, however, unimportantfor the computation of superamplitudes.In general, n -point tree-level scattering amplitudes can be written as follows [24], A (0) n = i δ (4) ( P ni =1 λ i ˜ λ i ) δ (8) ( P ni =1 λ i η Ai ) h ih i · · · h n i n − X k =0 P kn , (2.22)where P kn are polynomials in the Grassmann variables η i of degree 4 k . Invariance under R -symmetry implies that P kn are invariant under SU (4) rotations of the Grassmann vari-ables η A . The lowest-order term in the η expansion has Grassmann weight 8, while thehighest-order term has Grassmann weight 4 n −
8. CPT conjugation exchanges weight 4 k + 8with weight 4 n − k −
8. The k = 0 term in eq. (2.22) has P n = 1 and contains all the MHVamplitudes. The NMHV amplitudes are contained in the k = 1 term. Similarly to equation(2.19) and for the same reason, equation (2.22) is expected to hold to all orders in pertur-bation theory. Higher-order corrections can alter only the coefficients of the polynomials P kn , i.e. the component amplitudes. Throughout the paper, four-fold bosonic momentum-conserving delta functions will appear, products of delta functions over the four componentswhose indices (a vector index µ or a pair ( α, ˙ α ) of spinorial indices) we suppress. A variety offour-fold Grassmann delta functions, products of delta functions taken over the SU (4) index A , and eight-fold Grassmann delta functions, products of delta functions taken over a pairof a spinor index α and an SU (4) index A , will also appear. In these delta functions, we willsuppress the (bosonic) spinor index, but display the (Grassmann) SU (4) index explicitly.The tree-level MHV superamplitude was written down long ago by Nair [35], A (0) , MHV n = i δ (4) (cid:16)P ni =1 λ i ˜ λ i (cid:17) δ (8) (cid:0)P ni =1 λ i η Ai (cid:1) h i · · · h n i . (2.23)The MHV amplitude has an equally simple form in the conjugate superspace, whose coordi-nates are the conjugate spinors ˜ λ i and the Fourier-conjugate ˜ η of the Grassmann variables η . Fourier-transforming to the same superspace as the MHV amplitude implies [5] that theMHV superamplitude is A (0) , MHV n = i δ (4) ( P ni =1 λ i ˜ λ i )[12] · · · [ n Z d ω n Y i =1 δ (4) ( η Ai − ˜ λ ˙ αi ω A ˙ α ) . (2.24)11anifestly supersymmetric expressions for non-MHV amplitudes could be obtained [57]through a supersymmetric generalization of the MHV vertex expansion [58–60]. The expres-sions obtained this way do not a priori exhibit any special properties. DHKS presented [24]a special form for P , and showed that it enjoys an extended symmetry, so-called dualsuperconformal symmetry. Explicit expressions for all the P kn polynomials were given byDrummond and Henn [54], using a supersymmetric form [61, 62] of the Britto, Cachazo,Feng, and Witten (BCFW) on-shell recursion relations [7].On-shell superspace encodes the relations between amplitudes that are implied by su-persymmetry, but does not identify the basic, irreducible components from which all otherscan be obtained. Identifying such basic amplitudes, from which all others can be obtainedvia supersymmetry transformations (along with the required sequence of transformations)is in general a difficult problem. Not all corrections to the coefficients in the polynomials P kn are independent; as these coefficients are nothing but component amplitudes, they arerelated by supersymmetry Ward identities. Elvang, Freedman, and Kiermaier have provideda solution [63] to this class of questions.Apart from clarifying the structure of tree-level amplitudes, knowledge of tree-level su-peramplitudes allows us to perform manifestly-supercovariant higher-loop calculations usinggeneralized unitarity. The one-loop calculation of ref. [24] generalizes the result of ref. [2] forthe NMHV six-gluon amplitudes to a manifestly supersymmetric expression encompassingall possible external states. In refs. [6, 64] superamplitudes were used to evaluate the sumover all the particles crossing generalized unitarity cuts for n -point MHV amplitudes at anyloop order. In the section IV we will describe in detail the steps needed for evaluating two-and higher-loop superamplitudes for any number of external legs and Grassmann weight,and elucidate the subtleties that arise in such evaluations. C. The six-point NMHV superamplitude
As our focus in later sections will be on the two- and higher-loop six-point NMHV (su-per)amplitude, we first review and extend the supersymmetric results of ref. [24] for thetree-level and one-loop expressions for this amplitude.As is true for the component amplitudes, relations between rational functions of bosonicspinor products and Grassmann variables allow the tree-level superamplitude to be expressed12n several equivalent forms. We may identify a canonical form, which will also be useful forhigher-loop calculations, by starting from the ǫ -pole terms in the one-loop superamplitude.This superamplitude is given by the supersymmetrization [24] of eqs. (2.7), (2.12) and (2.13), A (1) , NMHV6 = a A (0) , MHV6 (cid:0) ( R + R ) W (1)1 + ( R + R ) W (1)2 + ( R + R ) W (1)3 + O ( ǫ ) (cid:1) , (2.25)where A (0) , MHV6 is the tree-level MHV superamplitude, the loop expansion parameter a isdefined in eq. (2.2) and the products A (0) , MHV R j,j +3 ,j +5 with j = 1 , . . . , A (0) , MHV R j,j +3 ,j +5 = δ (8) (cid:0)P λ i η Ai (cid:1) h j ( j +1) i h ( j +1) ( j +2) i [( j +3) ( j +4)] [( j +4) ( j +5)] (2.26) × δ (4) (cid:0) η Aj +3 [( j +4) ( j +5)] + η Aj +4 [( j +5) ( j +3)] + η Aj +5 [( j +3) ( j +4)] (cid:1) h j | K j +1 ,j +2 | ( j +3)] h ( j +2) | K j +3 ,j +4 | ( j +5)] s j,j +1 ,j +2 . This product is covariant under dual inversion, with the same weight as the tree-level MHVsuperamplitude. For generic momentum configurations (that is, away from soft or collinearconfigurations), the superfunctions R j,j +3 ,j +5 are thus invariant under dual superconformaltransformations.The functions W (1) i have identical poles in the dimensional regularization parameter ǫ ;this reflects the universality of infrared divergences. A canonical expression for the tree-levelsix-point NMHV superamplitude is then simply, A (0) , NMHV6 = 12 A (0) , MHV6 (cid:0) R + R + R + R + R + R (cid:1) . (2.27)The R invariants are not all independent; in the presence of the super-momentum conser-vation constraint they obey the linear six-term relation, A (0) , MHV6 (cid:0) R − R + R − R + R − R (cid:1) = 0 . (2.28)This relation, akin to relations derived from the Grassmannian formulation of tree-levelamplitudes [65], leads to two apparently different presentations of the six-point NMHVsuperamplitude: A (0) , NMHV6 = A (0) , MHV6 (cid:0) R + R + R (cid:1) = A (0) , MHV6 (cid:0) R + R + R (cid:1) . (2.29)13 proof of eq. (2.28) amounts to showing that the first expression in eq. (2.29) can bederived from the BCFW recursion relation [7] with a supersymmetric shift [62] while thesecond expression follows from the cyclic symmetry of the superamplitude. At higher loopsthe identity (2.28) is crucial for ensuring the consistency of unitarity cuts. It should alsoplay a role in reconstructing scattering amplitudes from their leading singularities [65].The six-point NMHV amplitude is special among NMHV amplitudes as it exhibits adiscrete invariance related to parity transformations. We will discuss this symmetry and itsconsequences in the following. A similar discussion generalizes to the 2 n -point N n − MHVamplitudes. As mentioned previously, (CPT) conjugation of superamplitudes amounts toFourier-transforming the Grassmann coordinates (reversing the helicities of all componentfields) and exchanging spinors and conjugate spinors, λ i ↔ ˜ λ i . It is easy to check that thissequence of transformations maps the products A (0) , MHV6 R ijk into themselves up to a cyclicpermutation by three units: A (0) , MHV6 R → A (0) , MHV6 R , etc. (2.30)This is the supersymmetric generalization of an obvious invariance of the six-gluon NMHVscattering amplitudes. Invariance of the six-point superamplitude under this transformationin turn requires that the functions W (1) i in equation (2.25) be invariant under conjugation.Apart from terms proportional to the sum of R invariants, the O ( ǫ ) part of the one-loop amplitude also contain terms which are proportional to differences of R invariants.They have been computed directly in a one-loop calculation [25], and their existence mayalso be inferred from the two-loop calculation we will describe in later sections. As suchdifferences are odd under conjugation, they must be accompanied by parity-odd (pseudo-scalar) functions f W (1) i . D. Dual Conformal Invariance and the Six-Point Superamplitude
As mentioned above, DHKS showed [24] that tree-level amplitudes are covariant, withweight ( − A (0) , MHV6 R ijk . To what extent does the symmetry extend to the full one-loopamplitude?The dual conformal and dual superconformal symmetries are only defined in four dimen-14ions. One possible extension is the notion of pseudo-conformality: were we to regulate theintegral functions W (1) i by off-shell continuation, they would become dual conformal invari-ant, as they are sums of box integrals with the appropriate prefactors. Additional evidencetowards a kind of dual conformal invariance comes from the observation [33, 34] that leadingsingularities are dual conformal invariant.We can do better than this. DHKS noticed [24] that the ratio of the six-point NMHV toMHV superamplitudes, each taken through one-loop order, is invariant under dual conformaltransformations. That is, the ratio is invariant under transformations that preserve thecross-ratios u = s s s s , u = s s s s , u = s s s s . (2.31)The ratio of superamplitudes is a natural quantity, as it is infrared finite.In gauge theories, the structure of infrared divergences in dimensional regularization isindependent of the helicity configuration [46, 47]. At one loop, for example, the pole termsare proportional to the tree amplitudes. This makes the ratio of any helicity amplitude tothe MHV amplitude infrared finite.The finiteness of such ratios makes it possible to take the four-dimensional limit, andto inquire about their properties under dual (super)conformal transformations. Of course,finiteness does not guarantee dual conformal invariance. Indeed, the relation between thesetwo properties has been investigated in ref. [5] with the conclusion that, in dimensionalregularization, there exist infrared-finite combinations of pseudo-conformal integrals whichare not dual conformal invariant.Explicit calculations show that such subtleties do not arise here and, through one-looporder, the six-point NMHV superamplitude has the factorized form [24] A NMHV6 = 12 A MHV6 h R (cid:16) aC (1)146 (cid:17) + cyclic + O ( a ) i , (2.32)with the functions C i,i +3 ,i +5 manifestly expressed in terms of the dual conformal ratios (2.31): C (1)146 = − ln u ln u + 12 X k =1 (ln u k ln u k +1 + Li (1 − u k )) − π O ( ǫ ) , etc. (2.33) This ratio is sensible because in chiral on-shell superspace any superamplitude is proportional to the super-momentum conservation constraint δ (8) ( P ni =1 λ i η Ai ), which contains the entire Grassmann-dependent fac-tor in the MHV amplitude. V (1) as defined in ref. [24] by − π /
6, due to differences in nor-malization of amplitudes and finite differences between the Wilson loop expression and theone-loop amplitude. It also differs in including O ( ǫ ) terms; its ǫ -independent part agreeswith the function V (1) defined in ref. [5].For completeness we record [1] the integral representation of the one-loop six-point MHVamplitude through O ( ǫ ): A (1) , MHV6 = A (0) , MHV6 M (1)6 , (2.34) M (1)6 = − X σ ∈S ∪S ∪S (cid:18) s s I ( σ ) + 12 ( s s − s s ) I ( σ ) (cid:19) . In writing eq. (2.33) we used the convention that u i +3 = u i . The ratio function is thusmanifestly dual conformal invariant through one loop. It does not have the full dual super-conformal invariance, dual supersymmetry being broken by a holomorphic anomaly [66].DHKS conjectured [24] that the main features of eq. (2.32) survive higher-loop corrections:that the six-point NMHV superamplitude may be factorized as A NMHV6 = 12 A MHV6 (cid:2) R NMHV6 + O ( ǫ ) (cid:3) ; (2.35)and that the functions R NMHV6 have no further ǫ dependence, are well-defined in four di-mensions and, to all loop orders, are dual conformal invariant. The conjecture does notspecify the structure of the O ( ǫ ) terms or of the spin factors that enter the functions R NMHV6 beyond one-loop level. At one-loop, the O ( ǫ ) terms are irrelevant to any ‘physical’ quantity.However, these terms will contribute nontrivially to both the divergent and finite parts ofthe O ( a ) terms in the product on the right-hand side of eq. (2.35). Our calculation willclarify the meaning of these one-loop terms for that part of the amplitude dependent onparity-even combinations of R invariants. We will show that they are determined by the O ( ǫ ) terms in the one-loop NMHV amplitude, which have been calculated recently by Dixonand Schabinger [25].Before proceeding to describe our calculation, we will discuss in the next section thestructure of our result as well as the expected properties of the resummed six-point NMHVamplitude. 16 IG. 2: Generalized cuts required to determine the two-loop NMHV amplitude: (a) the ‘double-pentagon’ cut (b) the ‘turtle’ cut (c) the ‘hexabox’ cut (d) the ‘flying-squirrel’ cut (e) the ‘rabbit-ears’ cut. Unlike the MHV calculation, all permutations of the external legs must be considered.
III. STRUCTURE OF THE SIX-POINT NMHV AMPLITUDE
In order to obtain the six-point NMHV amplitude to a given loop order, we must de-termine all spin factors that appear, and construct the functions of external momenta andcoupling multiplying each one of them. In the next section we will show explicitly that,through two-loop order and through O ( ǫ ), the R invariants are the only spin factors thatappear in the superamplitude. The transformation of the R invariants under conjugation(2.30) then implies that, through two-loop order, the superamplitude can be written asfollows, A NMHV6 = 12 A (0) , MHV6 h ( R + R ) W ( a ) + ( R + R ) W ( a ) + ( R + R ) W ( a )+ ( R − R ) f W ( a ) + ( R − R ) f W ( a ) + ( R − R ) f W ( a ) + O ( ǫ ) i , (3.1)17here the W i ( a ) are scalar functions and the f W i ( a ) are pseudoscalar functions. We presentthe calculation of the two-loop six-point NMHV superamplitude, computing explicitly theterms depending on parity-even combinations of R invariants. We will find that the four-dimensional cut-constructible part of the parity-even functions W (2) i can be expressed as asum of pseudo-conformal integrals. We will also confirm that, unlike their one-loop counter-parts, the pseudoscalar functions f W (2) i have nonvanishing divergent and finite parts in the ǫ expansion. We will not compute these functions explicitly, but the general infrared structureof gauge theories divergences requires that they have at most simple (1 /ǫ ) poles, as boththe tree and one-loop amplitudes [through O ( ǫ )] are free of such terms. In this section, wedescribe the expected general structure of the NMHV amplitude, and the structure of itscollinear limits. A. Beyond Two Loops
We expect the pseudo-conformality of the coefficient functions to continue to all looporders. To see this, consider a four-dimensional generalized unitarity cut that decomposesan L -loop superamplitude into a product of k tree-level superamplitudes A ( L ) n (cid:12)(cid:12)(cid:12) cut = Y A . . . A k . (3.2)As mentioned earlier and shown in [24], each superamplitude has weight ( −
1) under dualinversion. Because a cut propagator simply identifies the Grassmann variables and momentaof the legs that are sewn, it has weight (+2) under this transformation. Thus, the productabove together with the cut propagators has vanishing weight for the cut legs and weight( −
1) for the external legs. This implies that these cuts can all be saturated by cuts ofpseudo-conformal integrals.Unlike the scalar functions W (2) i , the pseudo-scalar functions f W (2) i are not uniquely de-fined. Indeed, the identity (2.28) implies that it is possible to uniformly add an arbitrarypseudoscalar function to the f W (2) i without affecting the value of the amplitude. In particu-lar, we could set any one of these functions to zero. This ambiguity can be partly eliminatedby requiring that the superamplitude be manifestly invariant under cyclic permutations ofexternal legs: f W (2) i = P f W (2) i − , (3.3)18 IG. 3: A cut of an L -loop six-point amplitude isolating an ( L − R invariant. where P is the operation of permutation to the right by one unit: P f [ k , k , k , k , k , k ] = f [ k , k , k , k , k , k ] . (3.4)The corresponding equation for the W (2) i functions, W (2) i = P W (2) i − , (3.5)follows from the symmetry of the superamplitude.Requiring cyclic symmetry does not completely fix the ambiguity in the pseudoscalarfunctions f W (2) i , as parity-odd cyclicly symmetric functions do exist. An example of such afunction is, f = ǫ f + ǫ f + ǫ f + ǫ f + ǫ f + ǫ f , (3.6)where f i are parity-even functions of external momenta k j related by the action of the shiftoperator f i = P f i − and ǫ ijmn = ǫ µνρσ k µi k νj k ρm k σn .The generalized-unitarity argument above does not reveal the complete set of spin factorsthat appear at higher loops in the six-point NMHV amplitude. The structure of leadingsingularities suggests [33] that new structures beyond the R invariants of one and two loopswill be generated at three loops for amplitudes with ten or more external legs, but that nonew structures will appear beyond that order. It also suggests that no new invariants shouldappear beyond two loops for amplitudes with seven or more, but fewer than ten, externallegs; and that no new invariants will appear beyond one loop for the six-point amplitude.We can, however, argue that the spin factors present at one and two loops will appearto all loop orders. As we will see in the next section, all tree-level R invariants appear indouble two-particle cuts in a channel carrying a three-particle invariant. Such a double cut,shown in fig. 2(a), isolates a tree-level four-point amplitude with no external legs attachedto it. An all-loop generalization of this cut is shown in fig. 3. The well-known property19f the four-point amplitude at any loop order, that its spin factor is the same as at treelevel, implies that this cut will generate exactly the same spin factors as at two loops. Thisargument extends trivially to all higher-loop contributions to the six-point amplitude thathave double two-particle cuts and isolate four- and five-point amplitudes inside them. Itcan be thought of as a direct superspace generalization of the box substitution rule [39].At three loops and beyond, however, it is easy to construct cuts that are outside this class.Such cut-based arguments thus cannot rule out spin factors beyond the R invariants seento date.Apart from terms containing such new spin factors which may start at three loops, theorganization of the six-point NMHV amplitude in (3.1) holds to all orders in perturbationtheory. It is therefore interesting to discuss the properties of the parity-even and parity-oddfunctions, W i ( a ) = 1 + aW (1) i + a W (2) i + . . . and f W i ( a ) = 1 + a f W (1) i + a f W (2) i + . . . (3.7)in equation (3.1) and of the finite functions C i,i +3 ,i +5 ( a ) = 1 + aC (1) i,i +3 ,i +5 + a C (2) i,i +3 ,i +5 + . . . (3.8)that appear in the ratio A NMHV6 /A MHV6 . The functions C i,i +3 ,i +5 ( a ) will not have definiteparity. This is due to their relation to linear combinations of functions with differing parityproperties, to wit ( W i ( a ) ± f W i ( a )), as well as to division by the MHV amplitude which doesnot have definite parity properties. For later convenience let us introduce the combinations C i ( a ) and e C i ( a ), 12 ( C i +3 ,i,i +2 + C i,i +3 ,i +5 ) ≡ C i ( a ) = W i ( a ) M ( a ) , (3.9)12 ( C i +3 ,i,i +2 − C i,i +3 ,i +5 ) ≡ e C i ( a ) = f W i ( a ) M ( a ) . (3.10)The properties of M ( a ) together with the universality of infrared divergences implies, thatthrough two loops, these functions have definite parity up to corrections that vanish in the ǫ → . To guarantee that the functions C i ( a ) and e C i ( a ) have definite parity to all orders in perturbation theoryit is necessary to divide only by the parity-even part of M ( a ). . Collinear Limits The scalar and pseudo-scalar functions W i ( a ) and f W i ( a ) have specific properties dictatedby the behavior of the amplitude in collinear limits [67]: A ( L )6 ( . . . , i λ i , ( i + 1) λ i +1 , . . . ) → X λ = ± L X s =0 Split ( s ) − λ ( z ; i λ i , ( i + 1) λ i +1 ) A ( L − s )5 ( . . . , k λ , . . . ) , (3.11)where k = k i + k i +1 and z is the collinear momentum fraction, k i ≃ zk . We can rewrite thisequation for the all-orders amplitude, A ( . . . , i λ i , ( i + 1) λ i +1 , . . . ) → X λ = ± Split − λ ( z ; i λ i , ( i + 1) λ i +1 ) A ( . . . , k λ , . . . ) . (3.12)The properties of C i ( a ) and e C i ( a ) are more intricate as they involve additional contributionsfrom M ( a ).We will find it easiest to discuss the collinear limits in components. Because W i and f W i do not depend on the precise helicity assignment to the external legs, it suffices to discussthe split-helicity configuration. In the three independent collinear limits, the spin factors B i in eqs. (2.4), (2.5) and (2.6) behave as follows:1 k B , → Split tree − (1 + , + , k − ) A (0)5 ( k + , + , − , − , − ) , B → k B , → Split tree+ (5 − , − , k + ) A (0)5 (1 + , + , + , − , k − ) , B → k B , → Split tree+ (3 + , − , k + ) A (0)5 (1 + , + , k − , − , − )+ Split tree − (3 + , − , k − ) A (0)5 (1 + , + , k + , − , − ) , B → e B i , which are contained in the parity-odd combinations of R invariants, are similar except that the relative sign between the twoterms in the 3 k k W + W → r − ( z ; 1 + , + ) E ( M MHV5 ( k, , , , f W + f W → r − ( z ; 1 + , + ) O ( M MHV5 ( k, , , , k W + W → r + ( z ; 1 − , − ) E ( M MHV5 (1 , , , , k )) f W + f W → r + ( z ; 5 − , − ) O ( M MHV5 (1 , , , , k ))3 k W + W → r + ( z ; 3 + , − ) E ( M MHV5 (1 , , k, , r − ( z ; 3 + , − ) E ( M MHV5 (1 , , k, , f W + f W → r + ( z ; 3 + , − ) O ( M MHV5 (1 , , k, , − r − ( z ; 3 + , − ) O ( M MHV5 (1 , , k, , r − λ ( z ; i λ i , ( i + 1) λ i +1 ) = Split − λ ( z ; i λ i , ( i + 1) λ i +1 )Split tree − ( z ; i λ i , ( i + 1) λ i +1 ) ; (3.15) M MHV5 and M MHV5 are the ratios of the resummed five-point MHV and MHV amplitudes totheir tree-level counterparts; and E and O denote projection operators onto the parity-evenand parity-odd components. Functions not explicitly mentioned are unconstrained.The collinear properties of the functions C i,i +3 ,i +5 and C i +3 ,i,i +2 can be easily found bycombining equations (3.9), (3.14) and the collinear properties of the MHV ratio M = A MHV6 /A (0) , MHV6 . In particular, they contain the Levi-Civita tensors necessary to transform,for example in the 1 k W i , which is similar in spirit to the BDS ansatz [8]:ln W i = ∞ X l =1 a l h f l ( ǫ ) W (1) i ( lǫ ) + C l + R ( l )6; i + O ( ǫ ) i . (3.16)The structure of infrared singularities and the collinear behavior require that O ( ǫ ) and O ( ǫ ) terms in the functions f l ( ǫ ) be the same as for the six-point MHV amplitude. Thefunctions R ( l )6; i , are similar in spirit to the remainder function R ( l )6 of the six-point MHVamplitude. They are closely related to the functions C i ( a ) introduced in equation (3.9): C i ( a ) = exp h γ K ( a ) (cid:16) W (1) i − M (1)6 (cid:17)i exp [ R i ( a ) − R ( a )] + O ( ǫ ) , (3.17)where γ K ( a ) is the cusp anomalous dimension and R i ( a ) = P ∞ l =2 a l R ( l )6; i , etc. A naturalconsequence of the conjecture that C i ( a ) are invariant under dual conformal transformationsis that the remainder-like functions R i ( a ) are also invariant. We will see that this is indeedso. The expectation [11, 68] that to leading order in the strong coupling limit, all amplitudeswith the same number of external legs are identical (or, equivalently, that lim a →∞ ln C i ( a ) = O ( a ) rather than O ( √ a )) predicts a simple relation between the remainder functions R i and the MHV remainder function R to this order. Indeed, using the one-loop relation W (1) i − M (1) = C (1) i (3.18)22nd the known value of the strong coupling expansion of the cusp anomaly [69–72], it followsthat R − R i C (1) i = √ λπ , (3.19)with C (1) i given in eq. (2.33). Using the numerical results presented in later sections onemay check that the weak-coupling expansion of the ratio appearing on the left-hand sidedepends on the spin factor labeled by i ; it seems therefore that a relation of this type mayhold only in the strong-coupling limit. C. Triple-collinear limits
Multi-collinear limits provide a richer set of constraints on amplitudes with at least sixexternal legs. Unlike the collinear limits discussed in the previous section, they probe thedetailed structure of the dual-conformal invariant functions unrelated to the infrared struc-ture of the amplitude. In the case of the six-point MHV amplitude, they provided a physicalinterpretation of the remainder function [19]. The most detailed limit we can consider withsix external legs involves three adjacent external momenta becoming collinear, k a = z P , k b = z P , k c = z P , z + z + z = 1 , ≤ z i ≤ , P → . (3.20)Let us understand what such limits imply about the six-point NMHV amplitude and, inparticular, about the remainder-like functions R i ( a ).An L -loop n -point amplitude factorizes as follows [67]: A ( L ) n ( k , . . . , k n − , k n − , k n ) X λ = ± L X s =0 A ( L − s ) n ( k , . . . , P λ ) Split ( s ) − λ ( k n − k n − k n ; P ) . (3.21)Taking into account parity and reflection symmetries, there are six independent triple-collinear splitting amplitudes [19]:Split + ( k + a k + b k + c ; P ) , (3.22)Split − λ P ( k λ a a k λ b b k λ c c ; P ) , λ a + λ b + λ c − λ P = 2 , (3.23)Split − λ P ( k λ a a k λ b b k λ c c ; P ) , λ a + λ b + λ c − λ P = 0 . (3.24)The first one (3.22) vanishes in any supersymmetric theory. The three triple-collinear split-ting amplitudes of the second type (3.23), an example of which is λ a = λ b = λ c = λ P = 1,23ppear in limits of MHV amplitudes. The N = 4 supersymmetry Ward identities for MHVamplitudes imply that their rescaled forms are all equal,Split ( l ) ∓ ( k ± a k + b k + c ; P )Split (0) ∓ ( k ± a k + b k + c ; P ∓ ) = Split ( l ) ∓ ( k + a k ± b k + c ; P )Split (0) ∓ ( k + a k ± b k + c ; P ) = r ( l ) S ( s ab s abc , s bc s abc , z , z ) . (3.25)These splitting amplitudes are relevant only for NMHV amplitudes with at least sevenexternal legs. They do not arise in the factorization of six-point amplitudes, because thefour-point amplitude entering the factorization (3.21) vanishes identically.The two splitting amplitudes of the third kind (3.24) arise only in limits of NMHVamplitudes and do not have a simple factorized form similar to (3.25) . They are howeverthe only splitting amplitudes that can appear in the triple-collinear limit of the six-pointNMHV amplitude.As is true for the tree-level NMHV amplitudes, the splitting amplitudes (3.24) have severaldifferent presentations related by potentially nontrivial spinor identities. A canonical one,that is useful for our purpose, is obtained from the triple-collinear limit of the six-pointtree-level amplitude in equations (2.14)-(2.16).As the functions W i are independent of the helicity assignment of the external legs, weagain discuss only the split helicity configuration. Up to conjugation and relabeling the onlynon-trivial limit is 2 k k
4. With the momentum fractions k = z P, k = z P, k = z P ,the spin factors B i become: b = B A (0)4 (1 + P + − − ) (1 − z ) √ z z z h i ( √ z [24] + √ z [34]) b = B A (0)4 (1 + P + − − )
7→ − ( √ z h i + √ z h i ) s h ih i ( √ z h i + √ z h i )+ [23] s [34]( √ z [24] + √ z [34])( √ z [23] + √ z [24]) (3.26) b = B A (0)4 (1 + P + − − ) z / √ z z (1 − z )[34]( √ z h i + √ z h i )+ ( z z ) / √ z h i ( √ z [23] + √ z [24]) . We omit a trivial dimensional dependence on s abc from the argument list of r ( l ) S . The spin-averaged absolute values squared of tree-level triple-collinear splitting amplitudes were computedin ref. [73]; without spin-averaging, in refs. [74]. The tree-level triple (and higher) collinear splittingamplitudes themselves were computed in ref. [75] using the MHV rules [76]. The one-loop correction tothe q → q ¯ QQ triple-collinear splitting amplitude in QCD was computed in ref. [77]. (0) − ( k +2 k +3 k − ; P ) = 12 ( b + b + b ) . (3.27)Thus, while these splitting amplitudes do not have a simple factorized form similar to thatfor splitting amplitudes of the second type (3.25), we see that the structure of the six-pointamplitude (3.1) implies that to this order each component b i is dressed at higher loops byscalar functions of momenta,Split − ( k +2 k +3 k − ; P ) = 12 ( b w ( a ) + b w ( a ) + b w ( a )) . (3.28)The parity-odd spin factors e B i also have nontrivial triple-collinear limits. Their coefficients f W i , though, must contain Levi-Civita tensors and thus naively vanish in this limit. Thetriple-collinear limits of additional spin factors that may appear beyond two loop ordermust be considered separately.As was true for the limit discussed in ref. [19], none of the conformal cross-ratios (2.31)vanish as 2 k k
4; they become¯ u = z z (1 − z )(1 − z ) , ¯ u = s s − z , ¯ u = s s − z . (3.29)Thanks to their expected dual conformal invariance (which we will confirm in later sections),the remainder-like functions R i ( a ) retain their complete kinematic content, and may beread off the two-loop triple-collinear splitting amplitude (3.24) by subtracting the triple-collinear limit of the two-loop iteration of the one-loop functions W (1) i . IV. CONSTRUCTING THE EVEN PART OF THE TWO-LOOP AMPLITUDE
We will construct the even part of the two-loop six-point NMHV amplitude using a super-space form [5, 6] of the generalized unitarity method [1–4, 21, 22, 78]. On general grounds,the result will be expressed as a sum of planar two-loop Feynman integrals with coeffi-cients that are rational functions of the spinor variables. At this order, one-loop calculationssuggest that it is possible to exclude integrals with triangle or bubble subintegrals.Similarly to the two-loop MHV amplitude, we will find that neither W (2) i nor f W (2) i can becompletely determined by four-dimensional cuts. Rather, they receive both divergent andfinite nontrivial contributions from integrals whose integrand is proportional to the ( − ǫ )25omponents of the loop momenta. It is quite nontrivial that these latter contributions canbe organized in terms of the same R invariants as the four-dimensional cut-constructibleterms.The generalized cuts that determine the amplitude are then the ones shown in fig. 2,which are the same ones that determine the MHV amplitude [19]. Unlike the calculation ofthe MHV amplitude, however, here it is necessary to evaluate cuts with all external helicityconfigurations, as each yields information about different spin factors.In any supersymmetric theory the improved power-counting ensures that at one-looporder and through O ( ǫ ) all terms can be detected in four-dimensional cuts. Beyond oneloop this is no longer true generically; for example, the six-point MHV amplitude at twoloops receives nontrivial contributions from integrals whose integrand vanishes identicallywhen evaluated in four dimensions. Four-point amplitudes in the N = 4 SYM theory are anexception: through five loops they appear to be determined solely by four-dimensional cuts.We therefore decompose the functions W (2) i in eqs. (3.1) and (3.7) into a four-dimensionalcut-constructible part and a part that requires D -dimensional calculations, W (2) i = W (2) ,D =4 i + W (2) ,µi . (4.1)For the former, powerful helicity and supersymmetry methods can be employed. The latterpart of the amplitude is determined by comparing the result of D -dimensional and four-dimensional calculations and is expressed in terms of “ µ -integrals” — nontrivial integralswhose integrand vanishes identically in four dimensions.While all cuts may be evaluated easily, separating their contributions to each one of thefunctions W (2) ,D =4 i is not always straightforward. As mentioned previously, (multiple) cutsin channels carrying three-particle invariants capture a single even and odd spin factor at atime and thus determine terms in a single W (2) ,D =4 i and f W (2) ,D =4 i , with the index i deter-mined by the helicity configuration of external legs. This is the case for cuts ( a ) and ( b )in fig. 2. In contrast, (multiple) cuts in channels carrying only two-particle invariants con-tribute simultaneously to several spin structures and thus to several W (2) ,D =4 i and f W (2) ,D =4 i functions. This feature is already present in the cut construction of the one-loop amplitude;in that case however, cuts in channels carrying three-particle invariants suffice to completelydetermine the amplitude [2]. The similarity between the expression for cut ( c ) of fig. 2 anda cut of the one-loop amplitude makes it possible to disentangle it. The cuts of fig. 2( d ) and26 e ) however seem intractable in a component approach.The component approach also fails to incorporate in a transparent way the constraintsimposed by supersymmetry. On-shell superspace provides the additional structure necessaryfor identifying the contributions of the remaining cuts to each of the W (2) ,D =4 i . We shalltherefore formulate the entire calculation of the four-dimensional cut-constructible part ofthe amplitude in on-shell superspace. After a brief overview of the structure of supercuts andof the techniques necessary to disentangle them, we will discuss cut ( a ), and then proceedto a more detailed analysis of the challenging cuts ( c ), ( d ) and ( e ). For the latter cuts weshall use a superspace generalization of the maximal cut method [39, 79]. A. Unitarity in Superspace: General Features and Techniques
Generalized cuts may be classified following the number of cut conditions they impose.The same is true for generalized supercuts. At L loops in four dimensions it is possibleto impose at most 4 L cut conditions; based on one- and two-loop information, it it likelythat their solutions generically form a discrete set. This type of cut has been consideredin the maximal-unitarity approach as well as in the leading-singularity approach. Maximalcuts, i.e. cuts with the maximal number of cut propagators, are typically insufficient tocompletely determine an amplitude. For example, at two loops one frequently encountersdouble box integrals, which cannot be detected by cutting eight propagators. Near-maximalcuts, obtained by successively relaxing cut condition in maximal cuts, provide an algorithmicway of identifying these contributions. Near-maximal cuts exhibit additional propagator-like singularities which are exploited in the leading singularity approach to reduce the one-parameter family of solutions to the cut conditions to a discrete set.The two-loop six-point NMHV amplitude can in principle be determined entirely fromthe iterated two-particle cuts shown in fig. 2. The Feynman integrals that contribute onlyto cuts ( c ), ( d ) and ( e ) are also detected by certain near-maximal cuts. We have used theminstead to check that the resulting amplitude correctly reproduces cuts ( c ), ( d ) and ( e ),supplemented by an additional cut condition isolating terms in one of the tree amplitudes.General supercuts are constructed [5, 6, 56] by multiplying together superamplitudes,identifying the η parameters of the lines that are sewn together and integrating over thecommon values of the internal η variables. The structure and properties of general supercuts27ave been analyzed in detail in ref. [6] where it was shown that, upon use of a supersymmetricgeneralization of the MHV vertex rules [76], their building blocks are generalized supercutsconstructed only out of MHV and MHV tree-level amplitudes.When evaluating a supercut one encounters the situation that on one side of the cut amomentum is outgoing and on the other side it is incoming. In order to write the tree-levelamplitude and in particular the argument of their delta functions, it is necessary to definethe spinors |− p i and |− p ] corresponding to the incoming momentum ( − p ). We use theanalytic continuation rule [56] that the change in sign of the momentum is realized by achange of sign of the holomorphic spinor p
7→ − p ↔ λ p
7→ − λ p , ˜ λ p +˜ λ p ; ↔ |− p i 7→ −| p i , |− p ]
7→ | p ] . (4.2)Let us discuss in detail the building blocks we require, supercuts constructed only out ofMHV and MHV tree-level amplitudes. Their evaluation requires the evaluation of integralsof products of delta functions with arguments linear in Grassmann parameters, see eqs. (2.23)and (2.24). For a p -particle cut of an N k MHV amplitude this product contains (8 + 4( k + p ))delta function factors of which (8 + 4 k ) remain upon integration. As discussed in refs. [5, 6,56], the integration over the internal η parameters realizes the sum over the states crossingthe (generalized) supercut. For any supercut, eight of these delta functions can always besingled out: they enforce the super-momentum conservation of the amplitude, δ (8) ( X i ∈E λ i η Ai ) (4.3)where E denotes the set of external lines. These delta functions may be thought of asthe supersymmetric generalization of the usual momentum conservation constraint. Theymay be extracted without carrying out any Grassmann integrations, by taking suitablelinear combinations of the arguments of all delta functions. If the supercut contains atleast one MHV superamplitude factor, the Jacobian of this transformation is unity. Theirextraction also makes manifest the invariance of the amplitude under half of the maximalsupersymmetry. Invariance under the other half of the supersymmetry, generated by¯ q ˙ αA = n X i =1 ˜ λ ˙ αi ∂∂η Ai , (4.4)is not manifest, but can in principle be checked at the level of the Grassmann integrand.28he delta functions (4.3) represent the complete Grassmann parameter dependence ofa supercut of an MHV amplitude. The Grassmann integrals simply yield the determinantof the system of linear equations which are the arguments of the other 4 p delta functions,where p is the number of cut lines [6].For cuts of an N k MHV amplitude there is a certain amount of freedom in evaluating theinternal Grassmann integrals. In general, however, the resulting 4 k delta functions havemany undesirable features. The essential ones are that (1) their arguments may dependon loop momenta (if the cut conditions do not completely freeze the momentum integrals)and (2) they may not make the symmetries of the amplitude manifest. We wish to expressthese Grassmann delta functions in terms of structures that appeared at lower-loop order; inthe case of the six-point NMHV amplitude; these are the dual superconformal R invariants.This is a non-trivial operation, and we have but a limited set of tools available.Given a set of 4 k delta functions k Y i =1 δ (4) ( e i ( η, λ )) , (4.5)it may be possible to construct linear combinations of their arguments M ij ( λ ) e j ( η, λ ) whichfactorize into products of the desired combinations of spinors and Grassmann variables uponuse of momentum and super-momentum conservation, cut conditions, and the fact that aGrassmann delta function equals its argument. For k = 1, which is the case of interest tous, no linear combinations can be constructed.A possible strategy for eliminating the dependence of the delta functions on loop momentais to make use of the fact that a Grassmann delta function equals its argument. Thisobservation replaces a cut carrying a Grassmann delta function with a sum of cuts of tensorintegrals with Grassmann-valued coefficients. Albeit nontrivial due to their high rank, thetensor integrals may then be reduced following the standard strategy of integral reduction.While indeed successful in eliminating the loop momentum dependence from the Grassmanndelta functions, this strategy is likely to lead to rather unwieldy expressions. We will notpursue this direction.An alternate approach to reorganizing Grassmann delta functions is to use the Lagrangeinterpolation formula, which is most efficient when applied to next-to-maximal cuts, whichimpose (4 L −
1) on-shell conditions. Let y be the variable that parametrizes the solutionto these cut conditions. The product of the 4 k Grassmann delta functions is then just a29olynomial P d ( y ) of degree d = 4 k with Grassmann-valued coefficients. Any such polynomialmay be written as P d ( y ) = d +1 X i =1 d +1 Y j =1 j = i y − y j y i − y j P d ( y i ) , (4.6)where the values y i are arbitrary. This equation simply encodes the fact that a polynomialof degree d is determined by its values at d + 1 points.Choosing the points y i can be regarded as freezing the momentum component unfixedby the cut condition; from this perspective it is akin to the leading-singularity methodwhich uses additional cut-like conditions for the same purpose. The Lagrange interpolationformula (4.6) provides a different strategy, as the points y i need not be chosen followingthe leading-singularity prescription. If the two approaches are to agree, the residue of theleading singularity must be proportional to the Grassmann delta functions appearing in dualsuperconformal invariants. Evidence that this is indeed true has been presented in ref. [33].In general, however, in order to use the interpolation formula (4.6), there must exist more y i such that P d ( y i ) is (proportional to) a dual superconformal invariant than are given bythe leading-singularity approach.In the next subsection we will use this strategy to analyze certain seven-particle cutsof the six-point two-loop NMHV superamplitude. As we will see, with judiciously chosenpoints y i it is possible to have P ( y i ) be proportional to the delta functions appearing in the R dual superconformal invariants.Because of the arbitrariness in the choice of the y i , the decomposition of P d ( y ) in a linearcombination of “good” Grassmann delta functions, such as the delta functions appearingin the dual superconformal invariants, is not unique. This signals the existence of linearrelations between the dual superconformal invariants. For six-point amplitudes, an identityarising this way is eq. (2.28), which was already required for the consistency of the variouspossible presentations of the tree-level amplitude. It is conceivable that at higher pointsand/or higher loops, new relations arise, beyond those that can be obtained from tree-levelconsiderations. 30 p H p + k L p H p + k L p H p + k L p H p + k L qp H q + k L H p + k L qp H q + k L H p + k L
13 14 15 p H p + k L FIG. 4: Two-loop topologies entering the 2-loop 6-point amplitudes. The arrow on the externalline indicates leg number 1.
B. Supercut Example: the Double-Pentagon Cut
Let us illustrate the general strategy outlined in the previous section, by examining insome detail two cuts that are essential for the construction of the six-point NMHV super-amplitude. We begin with the ‘double-pentagon’ cut, shown in fig. 2(a), which isolates thedouble-pentagon integrals I (12) and I (13) (shown in fig. 4) from a wide class of other integrals(hence its name).This cut provides two distinct contributions to the coefficients of the NMHV amplitude,depending on which of the two five-point tree-level factors is an MHV or an MHV super-amplitude. Each of the contributions is closed under supersymmetry transformations, so wewill call them supersectors. (They were called “holomorphicity configurations” in ref. [6].)31
23 456 l l l l (i)
123 456 l l l l (ii) FIG. 5: The two contributions to the ‘double-pentagon’ supercut 2(a). The circled + and − denoteMHV and MHV superamplitudes, respectively. The middle amplitude may be chosen to be eitherof MHV or of MHV type. Here we choose to present it as an MHV superamplitude. These two supersectors are shown in fig. 5; their values are, C dp Z d η l d η l d η l d η l d ω × δ (8) ( q A + λ l η Al + λ l η Al ) h ih ih l ih l l ih l i δ (8) ( λ l η Al + λ l η Al + λ l η Al + λ l η Al ) h l l ih l l ih l l ih l l i (4.7) × δ (4) ( η Al − ˜ λ ˙ αl ω A ˙ α ) δ (4) ( η Al − ˜ λ ˙ αl ω A ˙ α )[45][56][6 l ][ l l ][ l Y i =4 δ (4) ( η Ai − ˜ λ ˙ αi ω A ˙ α )and C dp Z d η l d η l d η l d η l d ω × δ (8) ( q A + λ l η Al + λ l η Al ) h ih ih l ih l l ih l i δ (8) ( λ l η Al + λ l η Al + λ l η Al + λ l η Al ) h l l ih l l ih l l ih l l i (4.8) × δ (4) ( η Al − ˜ λ ˙ αl ω A ˙ α ) δ (4) ( η Al − ˜ λ ˙ αl ω A ˙ α )[12][23][3 l ][ l l ][ l Y i =1 δ (4) ( η Ai − ˜ λ ˙ αi ω A ˙ α )where q αA = X i =1 λ αi η Ai and q αA = X i =4 λ αi η Ai . (4.9)By taking appropriate linear combinations of the arguments of the delta functions in theseequations it is easy to extract the overall super-momentum conservation delta function, δ (8) ( q A + q A ). Carrying out the Grassmann integrals we then find C dp A (0) , MHV R s h | | h | | h l l i ( h l ih l l ih l i ) ( h l l ih l l ih l l ih l l i ) ([6 l ][ l l ][ l , (4.10) C dp A (0) , MHV R s h | | h | | h l l i ( h l ih l l ih l i ) ( h l l ih l l ih l l ih l l i ) ([3 l ][ l l ][ l . (4.11)32e can reorganize the contributions into even and odd components, C dp = C dp C dp A (0) , MHV C dp+ ( R + R ) + A (0) , MHV C dp − ( R − R ) . (4.12)The two functions of vanishing weight in equations (4.10) and (4.11) may be identifiedas the contribution of gluon intermediate states in a component approach. They can be de-composed by standard means, by reconstructing propagators and organizing the numeratorinto a single trace . For example, C dp+ = s ( h l l ih l l ih l l ih l l i ) (cid:18) h | | h | | h l l i ( h l ih l l ih l i ) ([6 l ][ l l ][ l h | | h | | h l l i ( h l ih l l ih l i ) ([3 l ][ l l ][ l (cid:19) = 14 (cid:20) s ( s s − s s )( l + k ) ( l + l ) ( l + k ) + s s ( l + k ) ( l + l ) ( l + k ) + s s ( l + k ) ( l + l ) ( l + k ) + s ( s s − s s )( l + k ) ( l + l ) ( l + k ) + s s s ( k − l ) ( l + k ) ( l + k ) ( l + l ) ( l + k ) + s s s ( k − l ) ( l + k ) ( l + k ) ( l + l ) ( l + k ) + s s s ( k − l ) ( l + k ) ( l + l ) ( l + k ) ( l + k ) + s s s ( k − l ) ( l + k ) ( l + l ) ( l + k ) ( l + k ) + 1( l + k ) ( l + k ) ( l + l ) ( l + k ) ( l + k ) × s (cid:16) ( s s − s s )( k − l ) ( k − l ) + s s ( k − l ) ( k − l ) + s s ( k − l ) ( k − l ) + ( s s − s s )( k − l ) ( k − l ) (cid:17) + 1( l + k ) ( l + k ) ( l + k ) ( l + k ) (4.13) × (cid:16) s s s s − s (cid:0) s s s + s s s + s s s − s s s (cid:1)(cid:17)(cid:21) From this expression we can easily read off the coefficients of all integrals in fig. 4 that havea double cut in the s channel. Some integrals appear multiple times, corresponding todifferent cyclic permutations of external legs that have such a cut. The numerator factorsin the expression above are precisely those required to render the integrals invariant underdual inversion. As we did not need to specify the helicity labels of the external legs, all cutswith this topology can be obtained by simple cyclic relabeling. This last step is important to avoid the appearance of parity-even terms which are a product of twoparity-odd factors. i) (ii) (iii) FIG. 6: Next-to-maximal cuts that detect the integrals not easily isolated by the iterated two-particle cuts: from left to right, next-to-maximal cuts for the ‘hexabox,’ ‘flying-squirrel,’ and‘rabbit-ears’ cuts of fig. 2(c), (d), and (e), respectively. (i) (ii) (iii) (iv)
FIG. 7: The four possible assignments of internal helicities for the next-to-maximal ‘flying-squirrel’cut of fig. 6(ii). The ‘ ⊖ ’ vertices denote three-point MHV amplitudes while the ‘ ⊕ ’ vertices denotethree-point MHV amplitudes. The ‘turtle’ cut shown in fig. 2( b ) can be computed in a similar way, and also contributesa lone R invariant. These two cuts determine the coefficients of all integral topologies infig. 4 except I (7) , I (14) , and I (15) . Other cuts are necessary to determine these contributions.An efficient strategy, which makes use of the results obtained from the double-pentagon cutof fig. 2( a ) and the turtle cut of fig. 2( b ), is to analyze the relevant next-to-maximal cutsand find the remaining integrals one at a time. C. Supercut Example: A Contribution to the Flying-Squirrel Cut
Following this strategy, we present one contribution to the next-to-maximal cut, shown infig. 6(ii), that imposes additional cut constraints beyond the ‘flying-squirrel’ cut of fig. 2( d ).It serves to isolate one of our target integrals, I (7) , and allows us to determine its coefficient.As explained above, we impose the additional cut conditions because of difficulties in or-ganizing the results for cuts like the ‘flying-squirrel’ cut in terms of dual superconformal R C Z d η l d η l d η l d η l d η q d η q d η q δ (4) ([1 q ] η Al + [ q l ] η A + [ l η Aq )[1 q ][ q l ][ l × δ (8) ( − λ q η Aq + λ η A + λ l η Al ) h q ih l ih l q i δ (8) ( − λ l η Al + λ η A − λ l η Al − λ q η Aq ) h l ih l ih l q ih q l i× δ (4) ([ l η Aq + [4 q ] η Al + [ q l ] η A )[ l q ][ q l ] δ (8) ( λ q η Aq + λ η A + λ l η Al ) h q ih l ih l q i× δ (8) ( λ η A − λ l η Al + λ q η Aq − λ l η Al ) h l ih l q ih q l ih l i . (4.14)The expected overall super-momentum conservation constraint may be extracted by addingthe arguments of all the eightfold delta functions δ (8) to the last such function, and thenusing momentum conservation to eliminate λ q η Aq and λ l η Al . These transformations haveunit Jacobian.The remaining Grassmann integrals can be computed easily; we obtain: N = [1 q ] δ (4) (cid:0) h q ih q l ih q l i [4 q ] η A + h q l ih q ih q l i [4 q ] η A + h q l ih q l ih q l i [ q l ] η A + h q l ih q l ih l i [ l η A (cid:1) . (4.15)The contribution from the supersector in fig. 7(ii) is obtained by dividing N by the explicitdenominators in equation (4.14). This expression can be simplified in several different ways;we proceed by solving the cut conditions. The internal spinors (except for those associated35o q ) may be expressed conveniently in terms of two variables y and z : λ l = yλ − λ , ˜ λ l = ˜ λ , λ q = yλ ,λ l = − λ , ˜ λ l = ˜ λ + y ˜ λ , ˜ λ q = ˜ λ ,λ l = − zλ − λ , ˜ λ l = ˜ λ , λ q = zλ ,λ l = λ , ˜ λ l = − ˜ λ + z ˜ λ , ˜ λ q = ˜ λ . (4.16)The momentum q can be determined through momentum conservation; the condition thatit be on shell relates the two parameters y and z . These relations imply that all spinorproducts in eq. (4.15) that do not contain the holomorphic spinor | q i are monomials in y , z and spinor products of external momenta. The remaining holomorphic spinor products,which do contain | q i , can be converted into functions of y and external spinor products bymultiplying and dividing by [ q and using the identities, h l q i [ q
5] = −h y h , h l q i [ q
5] = h i [65] , (4.17) h l q i [ q
5] = h i ([16] + y [26]) , h l q i [ q
5] = h y h i [25] . The numerator factor N in eq. (4.15) is then, N = s s y z [ q δ (4) (cid:16) η A ( h − y h η A ( h − y h i [25])+ η A ( h − y h i [25]) + η A ( s − y h (cid:17) . (4.18)This expression is invariant, though not manifestly, under the action of the supersymmetrygenerators ¯ q = P ˜ λ∂ η .Overall super-momentum conservation provides the means to further simplify N . Bysubtracting P i η Ai h i (1 + 6)5] = 0, and adding P i yη Ai h i i [25] = 0 to the argument of thedelta function we find δ (8) ( X i λ i η i ) N = s s y z h i [ q δ (4) ( η A [56] + η A [61] + η A [15] + y ( η A [56] + η A [62] + η A [25])) . (4.19)This superspace expression has the two unwanted features already mentioned in sec-tion IV A: on the one hand, the argument of the delta function depends on the internalmomenta through the variable y ; on the other, it is not manifestly a function only of the36ame superspace structures as the tree-level amplitude (3.1). We wish to reorganize it interms of R invariants, and at the same time remove the dependence on internal momentafrom the arguments of the delta function by using the Lagrange interpolation formula (4.6)on the degree four polynomial, P ( y ) = δ (4) ( η A [56] + η A [61] + η A [15] + y ( η A [56] + η A [62] + η A [25]))= X i =1 5 Y j =1 j = i y − y j y i − y j P ( y i ) . (4.20)For this to be possible, as explained earlier it is necessary that there exist at least five values y i such that P ( y i ) is proportional to an R invariant. It turns out that there are at least sixsuch values: P (cid:18) h h (cid:19) = (cid:18) h i [56] h (cid:19) δ (4) ( η A [45] + η A [53] + η A [34]) ∝ R , (4.21a) P (cid:18) − [16][26] (cid:19) = (cid:18) [56][26] (cid:19) δ (4) ( η A [12] + η A [26] + η A [61]) ∝ R , (4.21b) P (cid:18) − h h (cid:19) = (cid:18) h i [56] h (cid:19) δ (4) ( η A [23] + η A [31] + η A [12]) ∝ R , (4.21c) P (cid:18) h ih i (cid:19) = (cid:18) h ih i (cid:19) δ (4) ( η A [56] + η A [64] + η A [45]) ∝ R , (4.21d) P (cid:18) s h (cid:19) = (cid:18) h i [56] h (cid:19) δ (4) ( η A [34] + η A [42] + η A [23]) ∝ R , (4.21e) P (0) = δ (4) ( η A [61] + η A [15] + η A [56]) ∝ R . (4.21f)For some of these cases, we have used overall super-momentum conservation constraint aswell as nontrivial spinor identities to transform the argument of δ (4) . As we will see shortly,only the first four values of y correspond to leading singularities.We can use the Lagrange interpolation formula for any five of the six special values { y , y , y , y , y , y } = (cid:26) h h , − [16][26] , − h h , h ih i , s h , (cid:27) . (4.22)Clearly, the decomposition obtained this way is not unique as there are six different possi-bilities. Let us denote them by L i , where i is the index of the missing root. In general wecan construct a five-parameter decomposition P ( y ) = X i =1 α i L i ( y ) , (4.23)37ith P α i = 1. The remaining parameters α i may be constrained by requiring that thesuperamplitude have additional manifest symmetries; for example, one may require that theparity of the superamplitude be manifest. We impose such a requirement in the following.All the presentations of the cut obtained for different possible choices of five values y i arephysically equivalent. However, they contain different R invariants; the existence of morethan five values y i is equivalent to the existence of nontrivial relations between R invari-ants. These relations hold only in the presence of the overall super-momentum conservationconstraint.We are now in position to assemble the result C −A (0) , MHV6 s s P ( y ) h i [56]( y h i − h i )( y h − h h y h y [26]) . (4.24)Inspecting the denominator of this expression, we see that the first four points y i in eq. (4.22)correspond to poles. They are in fact positions of leading singularities, as all of them arisefrom the [ q − factor in eq. (4.19) which is the Jacobian arising from solving the cutconditions.Choosing the five points y i to be { y , y , y , y , y } , we obtain, C s s s A (0) , MHV6 (cid:18) y ( y − y ) y ( y − y ) R + y ( y − y ) y ( y − y ) R − y ( y − y ) y ( y − y ) R − y ( y − y ) y ( y − y ) R + R (cid:19) . (4.25)Each of the denominators appearing in this expression may be identified with a propagatorevaluated on the kinematic configuration (4.16).Thus, the contribution of this supersector depends only on R invariants. We can decom-pose it in even and odd invariants ( R i,i +3 ,i +5 ± R i +3 ,i,i +2 ), following the form (3.1) of thesuperamplitude. To identify the part of C I (7) we need to subtract from it the contribution of all the other integrals in fig. 4,determined from the cuts of fig. 2(a) and (b). These cuts can contribute only terms propor-tional to the invariants R , R , R or R . Thus, we can conclude immediately that R arises solely as a coefficient of I (7) , whose coefficient must therefore be,12 A (0) , MHV6 s s s ( R + R ) . (4.26)38ndeed, it is intuitively clear that because of its topology, I (7) can appear in the coeffi-cient (4.25) only in terms that have no additional propagators.Carefully repeating this analysis for the other even invariants implies that the completecontribution of this supersector to the even part of I (7) ’s coefficient is, − A (0) , MHV6 ( R + R ) s ( s s − s s ) − A (0) , MHV6 ( R + R ) s ( s s − s s )+ 12 A (0) , MHV6 ( R + R ) s s s . (4.27)This conclusion must be checked against the other configurations in fig. 7. Fig. 7(iv) isthe parity conjugate of fig. 7(ii) and should therefore yield the same result for the even R invariants (and its negative for the odd-parity ones).The configurations in figs. 7(i) and 7(iii) and are parity conjugates of each other. Evalu-ating them following the same steps yields, C A (0) , MHV6 s s δ (4) ([61] η A + [15] η A + [56] η A ) h ih i [56][61] h h A (0) , MHV6 R s s s , (4.28) C A (0) , MHV6 s s δ (4) ([34] η A + [42] η A + [23] η A ) h ih i [23][34] h h A (0) , MHV6 R s s s . (4.29)Unlike the configuration in fig. 7(ii), the loop-momentum dependence here cancels completelyafter integration over the internal Grassmann variables. One may verify that evaluating(4.27) on the relevant internal kinematic configuration reproduces eqs. (4.28) and (4.29).The component of the flying-squirrel cut of fig. 2(d) that we evaluated shows that this cutcontributes to all two-loop scalar functions W (2) i . The same is true for the other cuts with thistopology but with cyclicly-permuted external legs. The sum over cyclic permutations maybe reorganized in terms of permutations of a single “even” spin coefficient R + R whichmultiplies three different integrals of the type I (7) with different assignments of external legs.We will use this presentation in the following section. V. THE TWO-LOOP SIX-POINT SUPERAMPLITUDE
We determined the four-dimensional cut-constructible even parts W (2) ,D =4 i of the two-loopsix-point NMHV superamplitude, W (2) , NMHV i = W (2) ,D =4 i + W (2) ,µi , (5.1)39s explained in the previous section, by analyzing the cuts shown in fig. 2 or next-to-maximalversions of them. We obtained an explicit expression for the remaining part, W (2) ,µi , cut-constructible only in D -dimensions, by comparing the results of D -dimensional and four-dimensional cut calculations. We have carried out the calculation without assuming a specific(possibly overcomplete) basis of two-loop integrals and found that the integrals listed infigs. 4 and 8 are necessary and sufficient to saturate the cuts of the even part of the amplitudethrough O ( ǫ ). For comparison, and because of changes in the labeling of these integralswith respect to the original calculations of the two-loop six-point MHV amplitude [19], wealso present it in our labeling. qp × µ p · µ q × µ p p
17 18
FIG. 8: µ -integrals entering the 2-loop 6-point amplitudes. The arrow on the external line indicatesleg number 1. A. The NMHV amplitude
The four-dimensional cut-constructible even part of all six-point two-loop amplitudes isbuilt out of a sum of the 16 integrals shown in fig. 4 S (2) ,D =4 (1l) = 14 c I (1) ( ǫ ) + c I (2) ( ǫ ) + 12 c I (3) ( ǫ ) + 12 c I (4) ( ǫ ) + c I (5) ( ǫ ) + c I (6) ( ǫ )+ 14 (cid:0) c a P − I (7) ( ǫ ) + c b P − I (7) ( ǫ ) + c c I (7) ( ǫ ) (cid:1) + 12 c I (8) ( ǫ ) + c I (9) ( ǫ )+ c I (10) ( ǫ ) + c I (11) ( ǫ ) + 12 c I (12) ( ǫ ) + 12 c I (13) ( ǫ )+ 12 c I (14) ( ǫ ) + 12 c I (15) ( ǫ ) + c I ( ǫ ) . (5.2)The coefficients c i , which differ between the MHV and the NMHV amplitudes, are func-tions of external momenta and the numerical coefficients are symmetry factors reflecting thesymmetries of each integral under cyclic permutations of external legs.40he functions W (2) ,D =4 i are constructed by summing S (2) ,D =4 over the sets of permutations S i in eqs. (2.9), (2.10) and (2.11) that map each superinvariant ( R i +3 ,i,i +2 + R i,i + i,i +5 ) intoitself: W (2) ,D =4 i = 18 X σ ∈S i S (2) ,D =4 ( σ ) + O ( ǫ ) . (5.3)Of the overall factor of 1 /
8, a factor of 1 / c j in the identity permutation entering S (2) ,D =4 in eq. (5.2) are: c = − s s s + s s s c = 2 s s − s s s s − s s s s + 2 s s s s c = s ( s s − s s ) c = s s c = − s s s c = s s s c a = − s ( s s − s s ) c b = 2 s s s c c = − s ( s s − s s ) c = 0 c = s s s c = s s s c = − s s s c = − s ( s s − s s ) c = s s c = 2 s s c = 0 c = 2 s s s c = not necessary c = s (2 s s − s s )(5.4)In dimensional regularization, the six-point two-loop (and quite likely all higher-pointhigher-loop) amplitudes receive contributions from integrals — collectively referred to as“ µ -integrals” — whose integrand vanishes identically when evaluated in four dimensions.The integrals shown in fig. 8 are of this type, where µ p and µ q denote the ( − ǫ ) componentsof the loop momenta. As noted in ref. [19], the integral I (17) vanishes identically as ǫ →
0; wewill therefore ignore it in the following. To determine the contributions of such integrals wecompare the result of the four-dimensional cut calculation with that of the D -dimensionalcuts and find that the even part of the amplitude also contains the terms, W (2) ,µi = X σ ∈S i c ! X σ ∈S ∪S ∪S s I (18) ( σ ) . (5.5)The coefficients c j bear certain similarities to the corresponding coefficients in the MHVamplitude. 41 . The MHV amplitude For completeness, and because of differences of notation from ref. [19], we also present theintegrand of the even part of the MHV amplitude. The four-dimensional cut-constructiblepart is given by M (2) ,D =4 = 116 X σ ∈S ∪S ∪S S (2) ,D =4 ( σ ) + O ( ǫ ) (5.6)where the coefficients c j in the identity permutation are given by c = s (cid:0) s s s + s s s c = 2 s s + s ( s s − s s ) (cid:1) c = s ( s s − s s ) c = s s c = s ( s s − s s ) c = − s s s c a = s ( s s − s s ) c b = − s s s c c = s ( s s − s s ) c = 2 s ( s s − s s ) c = s s s c = s (2 s s − s s ) c = s s s c = s ( s s − s s ) c = − s s c = 0 c = 0 c = 0 c = − s s ( s s − s s ) c = 2 s ( s s s − s s s )+2 s ( s s s + s s s ) − s ( s s s + s s s ) (5.7)The µ -integral contribution is M (2) ,µ = 116 X σ ∈S ∪S ∪S (cid:20) c I (17) ( σ ) + 12 c I (18) ( σ ) (cid:21) . (5.8)As mentioned previously, I (17) starts at O ( ǫ ) [19] and thus does not contribute through O ( ǫ ). C. A Comparison of the MHV and NMHV Amplitudes
A direct inspection of the integrals in fig. 4 and of their coefficients in eq. (5.4) re-veals that the even part of the two-loop six-point NMHV amplitude is a sum of pseudo-conformal integrals. This is similar to the MHV amplitude, for which the four-dimensional42ut-constructible part has a similar property [19], as may also be seen by directly inspectingthe coefficients listed in eq. (5.7).This is perhaps not completely surprising in light of the argument presented in section IIIthat all four-dimensional cuts can be reproduced by cuts of pseudo-conformal integrals. Thisstructure does not guarantee, however, that the even part of the amplitude is dual conformalinvariant, even after infrared divergences are removed appropriately. We return to this pointin the next section.The structure of the NMHV amplitude is quite similar to that of the MHV amplitude,with only subtle differences in the values of the coefficients. Two of the integrals that didnot contribute to the MHV amplitude — I (14) ( ǫ ) and I (16) ( ǫ ) — enter in the NMHV ampli-tude with nonvanishing coefficient; similarly, a topology that exists in both amplitudes — I (1) ( ǫ ) — appears in the NMHV amplitude with an additional pseudo-conformal numerator.Moreover, an integral that contributes to the MHV amplitude — I (8) ( ǫ ) — disappears fromthe NMHV one. We note also that a perfectly valid integral — I (15) ( ǫ ) — appears in neitherthe MHV nor NMHV amplitudes. It would be interesting to understand the significance ofthis observation.The properties of the MHV and NMHV amplitudes differ from those observed in thefour-point amplitudes through five loops: • All pseudo-conformal integrals appear with relative weights of ± • An integral appears with coefficient zero if and only if the integral is unregulated aftertaking its external legs off shell and taking ǫ → • It has been proposed that the signs ± W (1) ,D =4 i and W (2) ,D =4 i contains only a sumover the permutations in the set S i while the functions M (1) ,D =4 and M (2) ,D =4 contain sumsover all twelve permutations S ∪ S ∪ S . 43 I. DUAL CONFORMAL INVARIANCE
The explicit calculation in section IV of the two-loop NMHV six-point superamplitudeshows that it indeed has the structure anticipated in section III. We obtained explicitintegral representations for the scalar functions W (2) i , summarized in the previous section.As it is the case with all massless theories in four dimensions, the amplitude is infrareddivergent; to examine the dual conformal properties of the amplitude it is necessary toisolate these divergences. Based on the universality of infrared singularities and their ex-ponentiation DHKS proposed [24] that these divergences be removed by simply dividing bythe MHV amplitude. This ratio (2.35) was conjectured to be dual conformally invariant.An alternative method was described in section III, see eq. (3.16).Since the functions W ( l ) i have a natural decomposition into four-dimensional and D -dimensional cut-constructible contributions, the functions C ( l ) i introduced in equation (3.9)inherit a similar decomposition W ( l ) i = W ( l ) ,D =4 i + W ( l ) ,µi −→ C ( l ) i = C ( l ) ,D =4 i + C ( l ) ,µi l = 1 , . (6.1)At one loop, the µ -integral contribution C (1) ,µi vanishes in the limit ǫ →
0. Because ofinfrared divergences, they nevertheless give rise to nontrivial contributions at two loops inthe ratio with the MHV amplitude. The functions C ( l ) i contain terms that vanish as ǫ → C (1) i ( ǫ ) = W (1) i ( ǫ ) − M (1) ( ǫ ) C (2) i ( ǫ ) = W (2) i ( ǫ ) − M (2) ( ǫ ) − M (1) ( ǫ )( W (1) i ( ǫ ) − M (1) ( ǫ )) . (6.2)As explained in sect. II C, in the limit ǫ → C (1) i ( ǫ ) reduces to dual conformal invariantfunctions very closely related to the V ( i ) defined in ref. [24]. The higher-order terms in ǫ were not considered in ref. [24]; we take them as determined by the amplitude through therelation (3.9) between W ( l ) i and C ( l ) i .The infrared-divergent terms in both W (1) i ( ǫ ) and M (1) ( ǫ ) have the usual formDiv (1)6 = − ǫ X j =1 ( − s j,j +1 ) − ǫ . (6.3)Thus, C (1) i ( ǫ ) is manifestly finite; moreover, the only divergent contribution arising from thelast term in C (2) i ( ǫ ) is due to the overall factor of M (1) ( ǫ ).44 . Terms Requiring D -dimensional Cuts At one loop, the µ -integrals, requiring consideration of D -dimensional cuts, yield onlyterms of O ( ǫ ) in both W (1) i ( ǫ ) and M (1) ( ǫ ). This allows us to isolate the µ -integral contri-bution in eq. (6.2): C (2) ,µi = W (2) ,µi − W (1) ,µi Div (1)6 − M (2) ,µ + M (1) ,µ Div (1)6 + O ( ǫ ) , (6.4)where we used the universality of the one-loop infrared divergences (6.3) and kept only theterms that have nontrivial divergent and finite parts. For example, we dropped the termsin eq. (6.2) coming from the finite part of the overall factor M (1) ( ǫ ) in the last term.The last two terms in the equation above contain information already available in theMHV amplitude. Indeed, this exact combination appears in the iteration of the µ -integralsfor this amplitude [19]: M (2) ,µ = M (1) ,µ Div (1)6 . (6.5)Thus, C (2) ,µi is given by, C (2) ,µi = W (2) ,µi − W (1) ,µi Div (1)6 , (6.6)with W (2) ,µi given by eq. (5.5).This expression for W (2) ,µi may be further simplified by making use of the special prop-erties of the hexabox integral discussed in section IV.A of ref. [19], in particular eq. (4.6): I (18) [ µ ] = − ǫ ( − s ) − − ǫ I hex [ µ ] . (6.7)Thus, W (2) ,µi can be expressed exactly in terms of one-loop integrals, albeit in six dimensions.Moreover, using the fact that the one-loop hexagon integral is invariant under cyclic per-mutations of external legs, W (2) ,µi can be expressed in terms of the massless six-dimensionalhexagon integral with a coefficient given by the universal divergent part of one-loop ampli-tudes: W (2) ,µi = −
112 Div (1)6 I hex [ µ ] X σ ∈S i s (2 s s − s s ) + O ( ǫ ) . (6.8)The four terms in each sum are in fact equal.The µ -integral contribution to the one-loop six-point NMHV amplitude is not yet avail-able in the literature. Information on its structure may be obtained by analyzing a two-particle cut of the µ -integral contribution to the two-loop NMHV superamplitude. Dixon45nd Schabinger [25] have evaluated such a cut directly; quite surprisingly, they find that itcan be organized in terms of the same R invariants as the four-dimensional cut-constructibleterms. The µ -integrals’ contribution to the even part of the one-loop six-point NMHV am-plitude is, W (1) ,µi = − I hex [ µ ] X σ ∈S i s (2 s s − s s ) . (6.9)Combining this with eqs. (6.8) and (6.6) immediately shows thatlim ǫ → C (2) ,µi = 0 . (6.10)In other words, the complete µ -integral contribution to the six-point two-loop NMHV ampli-tude is completely accounted for by extracting an overall factor of the MHV superamplitude.Through similar manipulations it is possible to show that the remainder-like functions R (2)6; i introduced in eq. (3.16) do not receive any µ -integral contributions. Indeed, directlyexpanding eq. (3.16) to O ( a ) we find that, R (2)6; i = W (2) i ( ǫ ) − (cid:20) (cid:16) W (1) i ( ǫ ) (cid:17) + f ( ǫ ) W (1) i (2 ǫ ) (cid:21) . (6.11)Identifying the µ -integral contributions to each of the terms on the right hand side and usingthe universality of infrared divergences implies that R (2) ,µ i = W (2) ,µi ( ǫ ) − Div (1)6 W (1) i ( ǫ ) = C (2) ,µi + O ( ǫ ) . (6.12)It therefore follows from equations (6.10) that R (2) ,µ i does not receive any finite µ -integralcontributions.The same, however, cannot be said about the µ -integral contribution to the odd part ofthe amplitude. Indeed, repeating the steps that lead to equations (6.2) we find that thecoefficients of the parity-odd quantities ( R i +3 ,i,i +2 − R i,i +3 ,i +5 ) are, e C (2) i = f W (2) i − Div (1)6 f W (1) i . (6.13)While the µ -integral contributions to f W (2) i are given in terms of the hexabox integral or,equivalently in terms of the six-dimensional hexagon integral, their contributions to f W (1) i are given in terms of a restricted set of the one-mass pentagon integrals [25]. This suggeststhat, for the odd part of the superamplitude, the µ integrals cannot be cleanly separatedfrom the four-dimensional cut-constructible terms.46 . Numerical Evaluation of the Amplitude In order to further analyze the properties of the two-loop six-point NMHV amplitude,we turn to a numerical evaluation of the two-loop integrals. Thanks to the results describedin the previous section, we may focus on the four-dimensional cut-constructible part of theamplitude. The task of evaluating the integrals is simplified substantially by the fact thatall of them have already been evaluated at several distinct kinematic points in [19]. We haveevaluated additional kinematic points using the package MB [45] and the same Mellin–Barnesparametrization of integrals that was used in the calculation of the MHV amplitude. Apartfrom testing the symmetry properties of the amplitude, this calculation also verifies theexpected universality of two-loop infrared divergences. Viewed differently, a successful testof the universality of the infrared divergences is a strong indication of the completeness ofthe cut construction described in previous sections.We choose Euclidean kinematics for all configurations of external momenta. As in thecalculation of the MHV amplitude, the symmetries of the momentum configuration, K (0) : s i,i +1 = − , s i,i +1 ,i +2 = − , (6.14)make it particularly useful, as all cyclic permutations or external legs yield the same valuefor all integrals. This implies that all functions W (2) ,D =4 i are equal for all i = 1 , ,
3. Usingthe values of the integrals collected in the Appendix B of [19] we find W D =4 i ( K (0) ) = 1 + a ( − ǫ + 5 . . ǫ + 8 . ǫ )+ a (cid:16) ǫ − . ǫ + 25 . ± . ǫ − . ± . (cid:17) + O ( a ) . (6.15)Where they are not explicitly included, the errors do not affect the last quoted digit. We haveused the error estimated reported by CUBA [81]. In general we found the errors to be reliable,giving an accurate measure of the number of trustworthy digits. In some contributions,however, we found them to be underestimated, invariably in the presence of small integralswith a fast-varying integrand. In such cases, when
CUBA reports a large χ , we take theaverage value of the integrals to be the central value and quote the variation of the integralunder changes of sampling points as the error estimate. The issue presumably involvesintegration regions missed because of special properties of the integrand.As discussed previously, the construction of the functions C (2) i requires keeping higherorders in the small- ǫ expansion of the one-loop amplitude. For the MHV amplitude, we use47he expression in terms of the iterated one-loop amplitude and the remainder function R (2)6 .For the point K (0) we find C i ( a, ǫ, K (0) ) = 1 + a (0 . . ǫ + 0 . ǫ ) (6.16)+ a (cid:16) − . ± . ǫ − (2 . ± . − R (2)6 ( K (0) ) (cid:17) + O ( a ) . We note that the residue of the simple pole in ǫ vanishes within errors, as it should. Wehave confirmed this property for all the other kinematic points .From equation (6.15) we can also find the value of the remainder-like functions R (2)6; i introduced in equation (3.16) at the point K (0) : R (2)6; i = − . ± . . (6.17)Similarly to the error quoted for C i ( a, ǫ, K (0) ), the error of R (2)6; i is completely inherited fromthat of W (2) i .We have evaluated the amplitude at another kinematic point (denoted by K (1) in [19])related to K (0) by dual conformal transformations as well as two other points related to eachother but unrelated to K (0) : K (1) : s = − . , s = − . , s = − . , s = − . ,s = − . , s = − . , s = − . , s = − . ,s = − . ,K (3) : s i,i +1 = − , s = − / , s = − / , s = − / ,K (6) : s = − , s = − , s = − , s = − / , s = − / , s = − / ,s i,i +1 ,i +2 = − , (6.18)In listing the kinematic points we attempted to preserve the notation for the points used inref. [19]. We have collected our results for the values of C (2) i and R (2)6; i in tables I and II. Thethree dual conformal ratios( u , u , u ) = (cid:16) s s s s , s s s s , s s s s (cid:17) (6.19)for the kinematic points are listed in the second column of these tables. We have also verified analytically the cancellation of infrared singular terms through O ( ǫ − ). The completecancellation of infrared-singular terms was shown analytically by G. Korchemsky (private communication). ABLE I: Comparison of conformally-related kinematic points. C (2) i are the finite parts of theratios C i = W i /M at two-loops. R (2)6 is the two-loop remainder function of the six-point MHVamplitude. kinematic pt. ( u , u , u ) C (2)1 + R (2)6 C (2)2 + R (2)6 C (2)3 + R (2)6 K (0) ( , , ) − . ± . − . ± . − . ± . K (1) ( , , ) − . ± . − . ± . − . ± . K (3) ( , , ) 14 . ± .
003 12 . ± .
004 11 . ± . K (6) ( , , ) 14 . ± .
078 12 . ± .
035 11 . ± . R (2)6; i at conformally-related kinematicpoints. kinematic pt. ( u , u , u ) R (2)6;1 R (2)6;2 R (2)6;3 K (0) ( , , ) − . ± . − . ± . − . ± . K (1) ( , , ) − . ± . − . ± . − . ± . K (3) ( , , ) 5 . ± .
003 4 . ± .
004 4 . ± . K (6) ( , , ) 5 . ± .
078 4 . ± .
035 4 . ± . As mentioned previously, we left the MHV remainder function R (2)6 unevaluated. Its dualconformal invariance [19] R (2)6 ( K (0) ) = R (2)6 ( K (1) ) R (2)6 ( K (3) ) = R (2)6 ( K (6) ) (6.20)implies that the equality within errors of the relevant entries of table I extends to an equalityof the functions C (2) i . Alternatively, we could have evaluated the remainder function from theanalytic expression found in ref. [82], the integral representation in ref. [83], or the simplifiedform in ref. [84]. The results we obtain thus suggest that C (2) i and R (2)6; i are functions solelyof the conformal cross-ratios, that is, that they are indeed invariant under dual conformaltransformations. 49 II. SUMMARY, CONCLUSIONS AND SOME OPEN QUESTIONS
The maximally supersymmetric gauge theory in four dimensions is an ideal testing groundfor probing the properties of gauge theories at both weak and strong coupling. The largedegree of symmetry makes perturbative calculations tractable to relatively high orders whileits string-theory dual provides powerful tools for understanding its strong-coupling behavior.Its hidden symmetries yield additional constraints that go beyond their initial connectionto the integrability of the dilatation operator of the theory.In this paper we have computed the parity-even part of the two-loop six-point NMHVamplitude using generalized unitarity in superspace. We showed that the result is invari-ant under dual conformal transformations, after removal of universal infrared divergences(including terms arising from O ( ǫ ) contributions at one loop, computed by Dixon and Sch-abinger [25]). The dual conformal invariant content may be organized in several differentways. The exponentiation of both the infrared divergences and of the collinear splitting am-plitudes suggest the introduction of certain remainder-like functions which, similarly to theremainder function for MHV amplitudes, are functions only of the conformal cross ratios.We have shown that, to all orders in perturbation theory, it should be possible to reconstructthe remainder-like functions by evaluating certain triple-collinear splitting amplitudes.Several interesting issues related to the calculation described here, and to the structureof the perturbative expansion of the theory and its strong coupling expansion remain to beclarified.Through the AdS/CFT correspondence, Alday and Maldacena [11] argued that, to leadingorder in their strong-coupling expansion, all planar scattering amplitudes with fixed numberof external legs are essentially identical up to perhaps a rational function of momenta andpolarization vectors. Our calculation and arguments show that the weak-coupling structureof the six point amplitude involves at least six different spin factors dressed with scalarand pseudo-scalar functions to all orders in the weak-coupling expansion. Reconciling thisstructure with the AdS/CFT considerations remains an important open problem, whichappears to require that, in the strong-coupling limit, the remainder-like functions introducedin section III have a very simple relation to the MHV remainder function.Inspired by strong-coupling considerations, several groups showed that MHV amplitudeshave a close relation to certain light-like polygonal Wilson loops order-by-order in weak-50oupling perturbation theory, first in explicit calculations [16–19], and very recently, castin a more general setting [20]. A similar relation for non-MHV amplitudes remains anopen question; our numerical results provide check points for future calculations in thisdirection. The Wilson-loop formulation of the six-point MHV amplitude led to the analyticevaluation [82–84] of the remainder function at two loops. It seems likely that a Wilson-loop formulation of NMHV amplitudes will allow a similar evaluation of the remainder-likefunctions characterizing this amplitude.A direct comparison of the integrands of the six-point MHV and NMHV amplitudes attwo loops reveals that certain integrals appear in one but not the other, while the contribut-ing integrals enter with numerical coefficients that do not conform with the effective rulesinferred from four-point amplitudes. Moreover, one perfectly valid pseudo-conformal inte-gral does not appear in either one of these amplitudes. It would be interesting to developa better understanding of this pattern of numerical coefficients. Evaluation of higher-pointNMHV amplitudes at two loops may help in this direction.In our calculation dual conformal invariance is obscured by the dimensional regulator;removal of infrared divergences is a crucial step in studying the dual conformal propertiesof scattering amplitudes. Using four and five-point amplitudes as testing ground, it wasshown [32, 85] that regulating the infrared divergences by a particular symmetry breaking ofthe gauge group makes dual conformal invariance more transparent. It would be interestingto repeat the calculation described in this paper as well as that of the two-loop six-pointMHV amplitude in this framework. Apart from a better understanding of dual conformalinvariance, such an endeavor would clarify the interpretation of µ -integrals as well as that ofthe parity-odd component of amplitudes. While we did not compute the parity-odd part ofthe six-point NMHV amplitude explicitly, its µ -integral contributions make it quite differentfrom the parity-odd terms in MHV amplitudes with up to six external legs.A very interesting question relates to symmetries of scattering amplitudes. Up to anoma-lies introduced by the regulator, it is expected that dual conformal transformations leavescattering amplitudes invariant to all orders in perturbation theory. Again apart fromanomalies due to the presence of a regulator, ordinary conformal invariance exhibits ad-ditional anomalies of a holomorphic type [86], related to singular momentum configurations,already at tree level. An appropriate definition of a generating function for superamplitudeswith variable number of external legs [87] allows this anomaly to be circumvented, and to51e realized on scattering amplitudes through the one-loop level. Korchemsky and Sokatchevhave recently given [34] a general construction of conformal and dual conformal invariants(and hence of Yangian invariants). Other approaches to the construction of Yangian invari-ants were discussed by Mason and Skinner [88] and by Drummond and Ferro [89]. Theseinvariants have expressions that were conjectured [65] by Arkani-Hamed et al. to representthe leading singularities of scattering amplitudes to all orders in perturbation theory. All-order leading singularities have been derived directly by Bullimore, Mason, and Skinner [33].While the form of subleading singularities is not yet clear, this structure suggests that itmay be possible to realize both symmetries (and their closure) at higher loops. The ex-tended algebra does not uniquely determine the S-matrix of N = 4 super-Yang-Mills theory,however [34]. Unraveling its constraints on scattering amplitudes should prove fruitful.The study of non-MHV amplitudes at higher loops is only in its early stages. Our explicitcalculation provides an example of such an amplitude. The structure of these amplitudesis substantially richer than that of MHV amplitudes. It seems likely that new and excitingproperties as well as new calculational techniques are waiting to be discovered. Acknowledgments
We are grateful to Zvi Bern, Lance Dixon, Gregory Korchemsky, Robert Schabinger,Emery Sokatchev, and Arkady Tseytlin for useful discussions. We are especially gratefulto Lance Dixon and Robert Schabinger for sharing their results on the µ -integral contri-bution to the one-loop six-point NMHV amplitude prior to publication. We thank MarcusSpradlin and Zvi Bern for help with integral numerics. This work was supported in partby the European Research Council under Advanced Investigator Grant ERC–AdG–228301(D. A. K.), the US Department of Energy under contracts DE–FG02–201390ER40577 (OJI)(R. R.) and DE–FG02–91ER40688 (C. V.), the US National Science Foundation underPHY–0608114 and PHY–0855356 (R. R.) and PHY–0643150 (C. V.) and the A. P. SloanFoundation (R. R.). We also thank Academic Technology Services at UCLA for computersupport. Several of the figures were generated using Jaxodraw [90] (based on Axodraw [91]).52
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