OPE for all Helicity Amplitudes
Benjamin Basso, Joao Caetano, Lucia Cordova, Amit Sever, Pedro Vieira
OOPE for all Helicity Amplitudes
Benjamin Basso , Jo˜ao Caetano , Luc´ıa C´ordova , Amit Sever and Pedro Vieira Laboratoire de Physique Th´eorique, ´Ecole Normale Sup´erieure, Paris 75005, France Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Physics and Astronomy & Guelph-Waterloo Physics Institute, University of Waterloo,Waterloo, Ontario N2L 3G1, Canada Mathematics Department, King’s College London, The Strand, London WC2R 2LS, UK Centro de F ´ ı sica do Porto, Departamento de F ´ ı sica e Astronomia, Faculdade de Ci ˆ e ncias da Universidadedo Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel
Abstract
We extend the Operator Product Expansion (OPE) for scattering amplitudes in planar N = 4 SYM to account for all possible helicities of the external states. This is done byconstructing a simple map between helicity configurations and so-called charged pentagontransitions . These OPE building blocks are generalizations of the bosonic pentagons enter-ing MHV amplitudes and they can be bootstrapped at finite coupling from the integrabledynamics of the color flux tube. A byproduct of our map is a simple realization of parity inthe super Wilson loop picture. a r X i v : . [ h e p - t h ] D ec ontents A.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20A.2 Pentagons and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22A.3 Parity Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Within the so-called pentagon approach for null polygonal Wilson loops in conformal gaugetheories, one breaks up a null polygon into much simpler building blocks called pentagontransitions P ( ψ | ψ (cid:48) ). These ones govern the transitions between two eigenstates of the colorflux tube – see figure 1. a – and provide a representation of the Wilson loop W n – or moreprecisely of the finite ratio of loops [1] depicted in figure 2 – in the form of an infinite sumover all OPE channels, W n = (cid:88) ψ i P (0 | ψ ) P ( ψ | ψ ) . . . P ( ψ n − | ψ n − ) P ( ψ n − | e (cid:80) j ( − E j τ j + ip j σ j + im j φ j ) , (1)with { τ i , σ i , φ i } a base of conformal cross ratios, which receive individually meaning of time,space and angle in the i ’th OPE channel [1, 2].What makes this decomposition extremely powerful in planar N = 4 SYM theory isthat all of its building blocks can be computed at any value of the coupling thanks tothe integrability of the underlying theory. Namely, the flux tube spectrum is under totalcontrol [3] and the pentagon transitions can be bootstrapped [1, 4–7] following (a slightlymodified version of) the standard form factor program for integrable theories.Through the celebrated duality between null polygonal Wilson loops and scattering am-plitudes [8, 9], the decomposition (1) also provides a fully non-perturbative representationof the so-called Maximal Helicity Violating (MHV) gluon scattering amplitudes in planar N = 4 SYM theory. 2 ψ (cid:48) P ( ψ | ψ (cid:48) ) jj + 2 j − j − j + 1middle
12 340 − n − n − n − n − n − ( a ) ( b ) Figure 1: a ) The pentagon transitions are the building blocks of null polygonal Wilson loops. Theyrepresent the transition ψ → ψ (cid:48) undergone by the flux-tube state as we move from one square tothe next in the OPE decomposition. This breaking into squares is univocally defined by specifyingthe middle (or inner dashed) edge of the pentagon to be Z middle ∝ (cid:104) j − , j, j + 2 , j − (cid:105) Z j +1 − (cid:104) j − , j, j + 2 , j + 1 (cid:105) Z j − . b ) In the OPE-friendly labelling of edges, adopted in this paper, the middleedge of the j -th pentagon ends on the j -th edge. As a result, the very bottom edge is edge − n −
2. The map between the OPE index j and the more commoncyclic index j cyc reads j cyc = − ( − j (2 j + 3) mod n . In this paper we will argue that a suitable generalization of the pentagon transitions intosuper or charged pentagon transitions allows one to describe all amplitudes, for any numberof external particles with arbitrary helicities and at any value of the ’t Hooft coupling.While the key ingredient in having an OPE expansion such as (1) is conformal symmetry,a central ingredient in the charged pentagon approach will be supersymmetry.The idea of charging the pentagons is not entirely new, and already appeared in [4]where certain charged transitions were introduced and successfully compared against N k MHVamplitudes. More recently, further charged transitions were bootstrapped and matched withamplitudes in [6, 12, 13].The aim of this paper is to complete this picture by proposing a simple map betweenall possible helicity amplitudes and all the ways charged pentagons can be patched togetherinto an OPE series like (1). An interesting outcome of this charged pentagons analysis is asimple proposal for how parity acts at the level of the super Wilson loop, which, as far as weare aware, was not known before.
In the dual Wilson loop picture, N k MHV amplitudes are computed by a super Wilson loopdecorated by adjoint fields inserted on the edges and cusps [10, 11]. It is this super loop thatwe want to describe within the pentagon approach.3t first, let us first ask ourselves what would be a natural extension of (1) that allows forsome regions of the loop to be charged due to the insertion of these extra fields. The minimalmodification one could envisage is to generalize the pentagon transitions to super pentagontransitions or charged transitions, in which P ( ψ | ψ (cid:48) ) stands as the bottom component. Asfor the N = 4 on-shell super field, a pentagon would naturally come in multiple of fivecomponents P = P + χ A P A + χ A χ B P AB + χ A χ B χ C P ABC + χ A χ B χ C χ D P ABCD , (2)where χ is a Grassmann parameter, A = 1 , , , R -charge index, and where, for sake ofclarity, we have suppressed the states ψ and ψ (cid:48) . With these charged transitions at hand, wecould now imagine building up charged polygons such as P A ◦ P A ≡ (cid:88) ψ P A (0 | ψ ) P A ( ψ | e − E τ + ... , P AB ◦ P ◦ P AB ≡ (cid:88) ψ ,ψ P AB (0 | ψ ) P ( ψ | ψ ) P AB ( ψ | e − E τ + ... , (3) P AB ◦ P CD ◦ P AB ◦ P CD ≡ (cid:88) ψ ,ψ ,ψ P AB (0 | ψ ) P CD ( ψ | ψ ) P AB ( ψ | ψ ) P CD ( ψ | e − E τ + ... and so on. Here, an upper index represents a contraction with an epsilon tensor. Namely, weuse P A = (cid:15) ABCD P BCD , P AB = (cid:15) ABCD P CD and P ABC = (cid:15) ABCD P D to compress the expressionsabove.The most obvious change with respect to the MHV case is that R -charge conservationnow forbids some of the processes which were previously allowed and vice-versa. For instance,in the creation amplitude P AB (0 | ... ) we can produce a scalar φ AB out of the vacuum, sincethis excitation has quantum numbers that match those of the charged pentagon. At thesame time, neutral states such as the vacuum or purely gluonic states – which appeared inthe non-charged transitions – can no longer be produced by this charged pentagon.What stays the same is that all these charged transitions can be bootstrapped usingintegrability – as much as their bosonic counterparts. The scalar charged transition P AB and the gluon charged transition P ABCD , for instance, already received analysis of this sortin [4, 6]. The fermonic charged transitions, P A and P ABC , were more recently constructedin [12, 13].The super pentagon hypothesis (2) and its OPE corollary (3) are the two main inputs inthe charged pentagon program for helicity amplitudes. In the rest of this section we presenta simple counting argument supporting the equivalence between super OPE series and superamplitudes.The important point is that not all the N k MHV amplitudes are independent. Becauseof supersymmetry, many of them get linked together by means of so-called SUSY Wardidentities. At given number n of particles, there is a basis of N ( k, n ) amplitudes in terms ofwhich one can linearly express all the remaining ones. Both were denoted by P ∗ in these works. N ( k, n ) independentamplitudes, was beautifully analyzed in [14]. As explained below, the very same countingapplies to inequivalent super OPE series like (3).Counting the number of super OPE series is relatively easy:At first, one notices that the R -charge of a polygon is always a multiple of four, as aconsequence of SU (4) symmetry. The first two cases in (3), for instance, involve chargedpentagons with a total of 4 units, as for NMHV amplitudes, while the last example in (3)has a total of 8 units of charge, and should thus be related to N MHV amplitudes.In the NMHV case, the amount of charge in each of the n − n − n − n − n − /
4! ways of distributingfour units of charge between the n − N (1 , n ) in [14], see discussion below (3.12) therein.This kind of partitions no longer enumerate all cases starting with N MHV amplitudes.For instance, there are three independent ways of charging all the four pentagons of anoctagon with two units of charge, P AB ◦ P CD ◦ P AB ◦ P CD , P AB ◦ P AB ◦ P CD ◦ P CD , P AB ◦ P CD ◦ P CD ◦ P AB , (4)with the last line in (3) being one of them. (We can understand this as coming from the threepossible irreducible representations in ⊗ or, equivalently, as the three inequivalent waysof forming singlets in ⊗ ⊗ ⊗ .) Therefore, to count the number of N MHV chargedpolygons we have to consider not only the number of ways of distributing eight units ofcharge within four pentagons but also to weight that counting by the number of inequivalentcontractions of all the R-charge indices. Remarkably, this counting is identical to the onefound in [14] based on analysis of the SUSY Ward identities. This is particularly obviouswhen looking at Table 1 in [14] where the number of independent N MHV componentsfor 8 and 9 particles is considered. In sum, our construction in (3) generates precisely N (2 ,
8) = 105 , N (2 ,
9) = 490 , . . . different N MHV objects, in perfect agreement with thenumber of independent components arising from the study of the SUSY Ward identities.It is quite amusing that the notation in [14] with a partition vector λ = [ λ , . . . , λ n − ]seems perfectly tailored to describe the charged pentagon approach where we have n − λ i ∈ { , , , , } . It also guarantees that the most general N k MHVcounting works the same for both amplitudes and OPE series, and concludes this analysis.The next step is to endow the charged pentagon construction with a precise dictionarybetween charged polygons and helicity configurations of scattering amplitudes. The weight 3 = S λ =[2 , , , in their table is precisely the one explained in our above discussion. ≡ Figure 2: We study the conformally invariant and finite ratio W introduced in [1]. It is obtained bydividing the expectation value of the super Wilson loop by all the pentagons in the decompositionand by multiplying it by all the middle squares. The twistors that define these smaller pentagonsare either the twistors of the original polygon (an heptagon in this figure) or the middle twistorsdescribed in figure 1.a (in the above figure there are three distinct middle twistors, for instance). A compact way of packaging together all helicity amplitudes is through a generating function,also known as the super Wilson loop [10, 11] W super = W MHV + η i η j η k η l W ( ijkl )NMHV + η i η j η k η l η m η n η o η p W ( ijkl )( mnop )N MHV + . . . (5)where W N k MHV is the N k MHV amplitude divided by the Parke-Taylor MHV factor. Here,the η ’s are the dual Grassmann variables [15, 16]. They transform in the fundamental of the SU (4) R-symmetry, as indicated by their upper index A = 1 , , ,
4, and are associated tothe edges of the polygon, indicated by the lower index i = − , , , . . . , n − street , this is the most natural labelling from the OPE viewpoint. It makes itparticularly simple to locate the j -th pentagon in the tessellation: it is the pentagon whosemiddle edge ends on edge j . The map between this labelling and the conventional cyclicordering is explained in the caption of figure 1.The super loop (5) has UV suppression factors associated to its cusps. One can find in theliterature several different ways of renormalizing the loop, such as to remove these factors.The one most commonly used is the ratio function
R ≡ W super /W MHV , first introducedin [16]. For our discussion, however, the OPE renormalization is better suited: it is obtainedby dividing the super loop (5) by all the pentagons in its decomposition and by multiplyingit by all the middle squares [4]
W ≡ W super / w with w ≡ (cid:32) n − (cid:89) i =1 (cid:104) W i ’th pentagon (cid:105) (cid:33) / (cid:32) n − (cid:89) i =1 (cid:104) W i ’th middle square (cid:105) (cid:33) , (6) These UV divergences are T-dual to the IR divergences of the on-shell amplitudes.
6s shown in figure 2. The ratio function R and the loop W are then easily found to berelated to each other by R = W / W MHV . They are essentially equivalent, being both finiteand conformally invariant functions of the η ’s and shape of the loop, but only R is cyclicinvariant. Our goal in this section is to find the map between the different ways of gluing the chargedtransitions together, as in (3), and the components of the super loop (6). Put differently,we would like to find a map between the η ’s and the χ ’s such that W in (6) also admits theexpansion W = P ◦ P ◦ · · · ◦ P + χ χ χ χ P ◦ P ◦ · · · ◦ P + χ χ χ χ P ◦ P ◦ · · · ◦ P + . . . (7)in terms of the χ ’s.There are two important properties of the super loop that will be relevant to our discus-sion.First, recall that an η is associated to an edge of the polygon while a χ is associated toa pentagon. As such, there are many more terms in the η -expansion (5) or (6) of the superloop than there are in the χ -expansion (7). This is no contradiction, however. The reasonis that the η -components are not all linearly independent, since, as mentioned before, theyare subject to SUSY Ward identities. On the contrary, the χ -components all have differentOPE interpretation and, in line with our previous discussion, should be viewed as defininga basis of independent components for the amplitudes. In other words, the map between χ -and η -components is not bijective if not modded out by the SUSY Ward identities. We canthen think of the χ -decomposition as a natural way of getting rid of SUSY redundancy.Second, the η -components of W are not ‘pure numbers’, since they carry weights under thelittle group; e.g., upon rescaling of the twistor Z → α Z the component W transformsas W → W /α . These helicity weights cancel against those of the η ’s, so that W is weight free in the end. In contrast, the components in (7), as well as the corresponding χ ’s, are taken to be weightless. With this choice, the χ -components coincide with the onespredicted from integrability with no additional weight factors.We now turn to the construction of the map. The question we should ask ourselves is: What does it mean to charge a pentagon transition?
Said differently, how do we move fromone pentagon-component to another in the χ decomposition of P in (2)? To find out, it helpsthinking of the χ ’s as fermionic coordinates of sort and recall how usual (meaning bosonic)variables are dealt within the OPE set up.The bosonic cross ratios are naturally associated with the symmetries of the middlesquares. Namely, we can think of any middle square as describing a transition between twoflux-tube states, one at its bottom and the other one at its top, as depicted in 3. a . Attachedto this square are three conformal symmetries that preserve its two sides (left and right).To move in the space of corresponding cross ratios ( τ, σ, φ ) we act on the bottom state withthese symmetries ψ bottom −→ e − Hτ + iP σ + iJφ ψ bottom . (8)7 bottom ψ top left right ψ bottom ψ top j j − j + 1( a ) ( b ) Figure 3: a ) Any square in the OPE decomposition stands for a transition from the state atits bottom ( ψ bottom ) to the state at its top ( ψ top ). This transition is generated by a conformalsymmetry of the right and left edges of that square (conjugate to the flux time τ ). b ) Similarly,the super pentagon P represents a transition from the state at its bottom to the state at its top.In the fermionic χ -directions, this transition is generated by a super-conformal symmetry of the( j − j -th and ( j + 1)-th edges in this figure. Equivalently, we could act with the inverse transformations on the state at the top ( ψ top ),since these are symmetries of the left and right sides sourcing the flux. In other words, theOPE family of Wilson loops is obtained by acting on all the twistors below each middlesquare with the conformal symmetries of that square.Similarly, to move in the space of ‘fermionic coordinates’ we should act with a supercharge.In contrast to the previous case, these are now associated to the pentagons in the OPEdecomposition. A pentagon transition represents the transition between two flux-tube states,one on the bottom square and the other on the top square – the transition being inducedfrom the shape of the pentagon. So what we should do is to find the supercharge thatpreserves the three sides of the pentagon sourcing the two fluxes, i.e., the sides j − j and j + 1 in figure 3. b , and act with it on the state at its bottom ( ψ bottom ) or, equivalently, withthe inverse symmetry on the state at its top ( ψ top ). There is precisely one chiral superchargethat does the job, as we now describe.Recall that we have 16 chiral supercharges at our disposal, that is, Q aA where A is an R -charge index and a is an SL (4) twistor index. By construction they annihilate the superloop W on which they act as [18] Q aA = n − (cid:88) i = − Z ai ∂∂η Ai with Q aA W = 0 . (9)By definition, for a given supercharge not to act on, say, the i -th side of the super loop,we need the coefficient of ∂/∂η Ai to vanish. This can be achieved by contracting the SL (4)index a with a co-twistor Y such that Y · Z i = 0. In our case, since we want Q to be asymmetry of the three sides of a pentagon, the co-twistor should be orthogonal to Z j − , Z j and Z j +1 . There is exactly one such co-twistor: Y j ≡ Z j − ∧ Z j ∧ Z j +1 . (10)8t is then straightforward to define the operator ∂/∂χ Aj that charges the j -th pentagon.It acts as Y j · Q A on the state ψ bottom entering the j -th pentagon from the bottom or,equivalently, on what have created this state. In other words, ∂/∂χ Aj is defined as Y j · Q A in (9) but with the summation restricted to edges lying below the j -th pentagon: ∂∂χ Aj ≡ j − ) j ( j ) j ( j + ) j j − (cid:88) i = − Y j · Z i ∂∂η Ai . (11)Alternatively we could act on the state ψ top at the top of the pentagon by restricting thesummation to edges lying above the j -th pentagon and flipping the overall sign. These twoprescriptions yield the same result since the two actions differ by Y j · Q A where Q A is thefull supercharge annihilating the super loop.The normalization factor multiplying the sum in (11) needs some explanation. It isintroduced to make ∂/∂χ Aj weight free. In other words, it is defined such as to remove theweight of the co-twistor Y j used to define our supercharge. In our notation, ( i ) j extracts theweight of the twistor Z i in the j -th pentagon. This operation is unambiguous once we requireit to be local with respect to the j -th pentagon, meaning that it should only make use of thefive twistors of this pentagon. Indeed, given a pentagon p with five twistors Z a , . . . , Z e , theunique conformally invariant combination carrying weight with respect to a is given by( a ) p = (cid:104) abcd (cid:105)(cid:104) cdea (cid:105)(cid:104) deab (cid:105)(cid:104) eabc (cid:105)(cid:104) bcde (cid:105) . (12)Uniqueness is very simple to understand. If another such expression existed, its ratio with(12) would be a conformal cross-ratio, which of course does not exist for a pentagon. A niceequivalent way of thinking of the weight (12) is as the NMHV tree level amplitude for thecorresponding pentagon, that is ( i ) − j = W ( iiii ) tree j -th (cid:68) . (13)(Stated like this, the idea of dividing out by such weights is not new, see discussion around(132) in [4].) Multiplying three such weights to make the normalization factor in (11), wewould get(( j − ) j ( j ) j ( j + ) j ) = (cid:10) Z j − , Z j +1 , Z t | j , Z j (cid:11) (cid:10) Z j , Z b | j , Z j − , Z j +1 (cid:11) (cid:10) Z j +1 , Z t | j , Z j , Z b | j (cid:11) (cid:10) Z b | j , Z j − , Z j +1 , Z t | j (cid:11) (cid:10) Z t | j , Z j , Z b | j , Z j − (cid:11) , (14)where Z t | j / Z b | j refer to the top/bottom twistors of the j -th pentagon respectively. Equiv-alently, Z t | j / Z b | j are the middle twistors of the ( j +1)-th/( j − (cid:104)P (cid:105) = (cid:20) (cid:104) Z , Z , Z , Z − (cid:105) ( ) ( ) ( ) (cid:21) W ( − , − , − , − = 1 . (15)9 a c ¯ ψ φ ψ v a c χ χ χ χ Figure 4: Leading OPE contribution to the NMHV octagon component P ◦ P ◦ P ◦ P = ∂∂χ ∂∂χ ∂∂χ ∂∂χ W . For this component, each of the four pentagons in the octagon decompositioncarries one unit of R -charge and fermion number. From the flux tube point of view, this correspondsto the sequence of transitions in equation (17). A second minor ambiguity comes from the fourth power in (12) or (13). Due to its presence,to extract any weight we need to compute a fourth root, giving rise to a Z ambiguity. Inpractice we start from a point where the right hand side of (14) is real and positive for any j and pick the positive fourth root when extracting the weight on the left. Then everything isreal and can be nicely matched against the integrability predictions. This seems reminiscentof the sort of positivity regions of [19]. It would be interesting to study the Z ambiguityfurther, and possibly establish a connection to the positivity constraints of [19]. As a check of our map (11) we consider an eight-leg scattering amplitude, i.e., an octagonor, equivalently, a sequence of four pentagons. For concreteness, we focus on the exampleof P ◦ P ◦ P ◦ P = ∂∂χ ∂∂χ ∂∂χ ∂∂χ W at tree level and evaluate it in terms of the nine OPEvariables { τ i , σ i , φ i } . At this order, the OPE ratio W coincides with the ratio function R and we can easily extract components of the latter from the package [30]. For large OPEtimes we find that P ◦ P ◦ P ◦ P = e − τ − iφ / × e − τ × e − τ + iφ / × f ( σ i ) + . . . (16)which is actually already a non-trivial check of our construction. Indeed, we have four chargedpentagons each of which injects one unit of R-charge and one unit of fermion number. Assuch, the lightest states that will flow in the three middle squares are a fermion ¯ ψ (withhelicity − /
2) in the first square, a scalar φ (with no helicity) in the second square andthe conjugate fermion ψ = ψ (with helicity +1 /
2) in the last middle square. In short,the leading process contributing to this amplitude should correspond to the sequence of10ransitions vacuum P −→ ¯ ψ P −→ φ P −→ ψ P −→ vacuum , (17)as represented in figure 4. The three exponential factors in (16) are in perfect agreementwith this expectation.Most importantly, the function f ( σ i ) should be given by the multiple Fourier transformof the sequence of pentagon transitions. It beautifully is. This and other similar checks – attree level and at loop level – will be the subject of a separate longer publication [7] whosemain goal will be to precisely confront the program advocated here against the availableperturbative data for non-MHV amplitudes. It is rather straightforward to invert the map (11) such as to obtain the ∂/∂η ’s in terms ofthe ∂/∂χ ’s. For that aim, it is convenient to put back the weights in (11) and define D ( j ) A ≡ ( j − ) j ( j ) j ( j + ) j ∂∂χ Aj = Y j · j − (cid:88) i = − Z i ∂∂η Ai . (18)Given the triangular nature of this map, charging the first few edges at the bottom is as easyas writting the first few D ’s explicitly. For the bottom edge, for instance, we immediatelyfind that D (1) A = Y · Z − ∂∂η A − ⇒ ∂∂η A − = D (1) A Y · Z − , (19)while taking this into account and moving to the following edge yields ∂∂η A = ( Y · Z − ) D (2) A − ( Y · Z − ) D (1) A ( Y · Z − )( Y · Z ) , (20)and so on.By following this recursive procedure we will eventually find that ∂/∂η j is given as alinear combination of D ( j +2) , D ( j +1) , . . . , D (1) . In plain words, it means that chargingthe edge j entails charging the entire sequence of pentagons lying all the way from thatspecific edge to the bottom of the polygon. The drawback is that it has be so even foran edge standing arbitrarily far away from the bottom of the polygon. This, however, isat odds with the locality of the OPE construction, in which a random pentagon in thedecomposition only talks to its neighbours (through the flux-tube state that they share) andhas little knowledge of how far it stands from the bottom. Besides, it introduces an artificialdiscrimination between bottom and top, despite the fact that our analysis could, at no cost,be run from the top. The way out is easy to find: the bottom tail of the inverse map is puremathematical illusion, or, put differently, the inverse map beautifully truncates such as tobecome manifestly top/bottom symmetric. 11 j − j +1 j +2 j +3 j − j − Figure 5: A remarkable feature of our construction is that the inverse map turns out to be local.Namely, charging edge j is done by charging the five pentagons touching this edge and these fivepentagons alone. We notice in particular that the two outermost pentagons in this neighbourhood,which are shown in green above, are touching the endpoints of edge j only. In sum, instead of a sum over j + 2 D ’s, what we find is that (for 3 ≤ j ≤ n − ∂/∂η j is given by the linear combination of the five neighboring pentagons only (see figure 5) ∂∂η Aj = (cid:104) Y j − , Y j − , Y j , Y j +1 (cid:105) D ( j +2) A + . . . + (cid:104) Y j − , Y j , Y j +1 , Y j +2 (cid:105) D ( j − A ( Y j − · Z j +1 ) ( Y j +1 · Z j − ) ( Y j − · Z j ) ( Y j +2 · Z j ) . (21)Mathematically, this relation originates from the five-term identity (cid:104) Y j − , Y j − , Y j , Y j +1 (cid:105) Y j +2 + . . . + (cid:104) Y j − , Y j , Y j +1 , Y j +2 (cid:105) Y j − = 0 , (22)which holds for any choice of five (co-)twistors and which simply follows from them havingfour components. Once we plug the definition (18) into the right hand side of (21), mostterms cancel out because of this identity. Those that survive are boundary terms and it isstraightforward to work them out in detail. They precisely lead to the single term in the lefthand side of (21). Actually, it is possible to interpret the inverse map (21) such that it also applies to thevery first edges of the polygon, like in (19) and (20), provided that we properly understandwhat we mean by Y , Y − , Y − and Y − . (These co-twistors will show up when using (21)for ∂/∂η , ∂/∂η , ∂/∂η and ∂/∂η − .) For this we can pretend that there are extra edges at To see that only the term proportional to ∂/∂η j survives it is useful to note that the orthogonalityrelations Y j − · Z j − = Y j − · Z j − = Y j · Z j − = 0 allow us to freely extend slightly the summation rangeof some of the five terms in (21). In turn, these relations follow trivially from the definition (10) of theco-twistors. Finally, to check the overall normalization of both sides in (21), it is convenient to use theidentity (cid:104) Y j − , Y j − , Y j , Y j +1 (cid:105) = ( Y j − · Z j +1 ) ( Y j +1 · Z j − ) ( Y j − · Z j ). Of course,for these bottom (or top) cases, it is easier to proceed recursively as in (19) and (20).This concludes our general discussion of the map. The proposals (21) and (11) are themain results of this paper.
A polygon with n edges has a top pentagon and a bottom pentagon, plus n − ∂/∂χ , is simply proportionalto ∂/∂η − , ∂∂χ = Y · Z − ( ) ( ) ( ) × ∂∂η − (23)which we can further simplify to (see e.g. (57) in the appendix for a thorough explanation) ∂∂χ = ( − ) × ∂∂η − . (24)In other words, up to a trivial factor which absorbs the weight in ∂/∂η − , charging a bottompentagon is the same as extracting components with η ’s at the very bottom of our polygon.Similarly, charging the top-most pentagon is equivalent to putting η ’s on the topmost edge.It could hardly be simpler. Explicitly, for any polygon, there are five NMHV componentswhich are easy to construct: P ◦ P ◦ · · · ◦ P ◦ P = w [4] W ( − , − , − , − , P ◦ P ◦ · · · ◦ P ◦ P = w [3] W ( − , − , − ,n − , P ◦ P ◦ · · · ◦ P ◦ P = w [2] W ( − , − ,n − ,n − , (25) P ◦ P ◦ · · · ◦ P ◦ P = w [1] W ( − ,n − ,n − ,n − , P ◦ P ◦ · · · ◦ P ◦ P = w [0] W ( n − ,n − ,n − ,n − . where w [ m ] ≡ (( − ) ) m (( n − ) n − ) − m .These are what we call the easy components . Morally speaking, from the first to the lastline, we can think of the easy components as inserting an F , ψ , φ , ¯ ψ , ¯ F excitation and theirconjugate at the very bottom and top of our polygon.For an hexagon we have only two pentagons and thus the easy components in (25) with n = 6 suffice to describe the NMHV hexagon, see figure 6. All other components can be We can simply define ( Y , Y − , Y − , Y − ) ≡ (cid:0) Y { , − , } , Y {∗ , , − } , Y {− , ∗ , ∗} , Y {∗ , ∗ , ∗} (cid:1) , with Y { i,j,k } ≡ Z i ∧ Z j ∧ Z k and Z ∗ being arbitrary twistors, which drop out of the final result. At the same time, we also set( D W , D − W , D − W , D − W ) = (0 , , , . − P ◦ P ( a ) ( b ) P ◦ P P ◦ P P ◦ P P ◦ P
Figure 6: ( a ) OPE friendly edge labelling used in this paper (big black outer numbers) versus themore conventional cyclic labelling (small red inner numbers) for the hexagon. ( b ) The five easycomponents of the NMHV hexagon. Each black square represents a dual Grassmann variable η .For the hexagon these five components provide a complete base for all NMHV amplitudes. trivially obtained by Ward identities. For example, we can use invariance under Y · Q = (cid:88) k Y · Z k ∂∂η k = Y · Z − ∂∂η − + Y · Z ∂∂η + Y · Z ∂∂η (26)to replace any component with an index associated to the edge 0 to a linear combination ofcomponents with η ’s associated to the top and bottom edges − W ( − , − , − , = α P ◦ P + β P ◦ P , (27)with α = − Y · Z − Y · Z (( − ) ) , β = − Y · Z Y · Z (( − ) ) ( ) . (28)Similarly, we can easily write down any other hexagon NMHV component in terms of theOPE basis. Of course, this is equivalent to using the general inverse map (21), worked outin the previous section.There are other components whose OPE expansions closely resemble those of the nice components (25). A notable example is the so-called cusp-to-cusp hexagon scalar component R hex − , , , and its heptagon counterpart R hep − , , , . Such components were extensively analyzedin the past using the OPE [4, 12, 21, 22]. What is nicest about them is their utter simplicityat tree level, being described by a simple scalar propagator from the bottom cusp ( − ,
0) tothe top cusp ( n − , n − τ and to any loop order. Both are described by Recall once again that we are using here a slightly unconventional labelling of the edges as indicated infigure 1.b; These same cusp-to-cusp component were denoted R and R in [4] and R in [21].
14 single scalar flux tube excitation. However, as soon as two-particle contributions kick in– in the sub-leading collinear terms – these components start differing. A similar story ispresent for all other components in the family (25). For example, gluonic components wereintensively studied in [6]. A simple tree-level gluonic example of this universality for theleading terms in the OPE was considered in detail in [23]. The hexagon fermonic component W ( − , − , − , was recently studied in [20]. Finally let us note that the weight factors showing up in (25) are not a novelty. Alreadyin [4] it was explained that to properly deal with weight free quantities we better remove theweight of each pentagon by dividing out by the corresponding charged counterpart, see (132)and surrounding discussion in [4]. Nevertheless, in practice, in all previous OPE studies ofsuper amplitudes, the weights ( − ) and ( n − ) n − of the bottom and top twistors withrespect to the corresponding pentagons were, for the most part, ignored. Sometimes thisis fine. For instance, if we are interested in amplitudes at loop level we can always dividethe ratio function by its tree level expression obtaining a weight free function of cross-ratioswhich we can unambiguously match with the OPE. Said differently, we can always normalizethe tree level result by hand such that it agrees with the leading terms in the OPE. Inparticular, for the purpose of comparing with the hexagon function program [24] and usingthe OPE to generate high loop order predictions, it is overkilling to carry these weightsaround. Moreover, with the choice of twistors in [4], such weights actually evaluate to 1which is one further reason why we never needed to take them into account.Having said all that, of course, to be mathematically rigorous, weight free quantities (25)are what we should always manipulate. In particular, for higher n -gons, and as soon as wealso charge middle pentagons, it is important to keep track of these weights to properly makecontact with the OPE predictions [7]. The charged pentagon construction provides us with a novel intuition about how to under-stand parity at the Wilson loop level.Recall that the action of parity on a scattering amplitude is very simple. It is a symmetryof the amplitude under which a positive helicity gluon transforms into a negative helicityone, a positive helicity fermion transforms into its negative helicity conjugate counterpartand finally a scalar excitation is trivially conjugated. All in all, this can be summarized inthe following nice relation [25] (cid:90) n − (cid:89) i = − d ˜ η i e (cid:80) n − i = − ¯˜ η i ˜ η i A [˜ η, λ, ˜ λ ] = A [¯˜ η, ˜ λ, λ ] . (29)However, the relation between amplitudes and super Wilson loops involves stripping outthe MHV tree-level factor along with going from the original amplitude ˜ η ’s to the Wilsonloop dual Grassman variables η ’s. Together, these operations obscure the action of parity This component was denoted W (1114) in the cyclic labelling in [20]. The convention for the labelling of the η ’s and ˜ η ’s varies quite a lot in the literature. Our notation hereis in line with [11] and [19] for example (modulo the non-cyclic labelling of the edges of course). χ -components – parity at the Wilson loop level is no more complicated than in theoriginal amplitude language. Precisely, we claim that our variables allow for a straightforwardanalogue of (29) in the Wilson loop picture as (cid:90) n − (cid:89) i =1 d χ i e (cid:80) n − i =1 ¯ χ i χ i R [ χ, Z ] = R [ ¯ χ, W ] , (30)where W j are Hodge’s dual momentum twistors [27]. The latter can be thought of as parityconjugate of the Z ’s and, up to an overall factor which drops out in (30), are given by W j ≡ Z j − ∧ Z j ∧ Z j +2 . (31)Note that this is nothing but the conventional definition of the dual twistor involving threeconsecutive edges; the shifts of 2 in the index are just an outcome or our labelling, seefigure 1.The general relation (30) is a generating function for all parity relations between N k NMHVcomponents and N n − − k MHV components, such as the relation P ◦ P = P ◦ P | Z → W (32)between two NMHV hexagon components, for instance, or the relation P ◦ P ◦ P = P ◦ P ◦ P | Z → W (33)relating NMHV and N MHV seven-point amplitudes. More precisely, to convert such identityinto a relation for ratio function components, it is suffices to divide both sides by W = P ◦ P ◦ P . After doing so, the same relation in (33) reads ∂∂χ ∂∂χ ∂∂χ ∂∂χ ∂∂χ ∂∂χ ∂∂χ ∂∂χ R heptagon N MHV = ∂∂χ ∂∂χ ∂∂χ ∂∂χ R heptagon NMHV (cid:12)(cid:12)(cid:12)(cid:12) Z → W (34)where the χ derivatives are given in terms of conventional η derivatives in (11). Anothermore extreme example following from (30) is the relation P ◦ P ◦ P = P ◦ P ◦ P| Z → W (35)which encodes the fact that for seven points N MHV is the same as MHV. It is straightfor-ward to generate more such relations by picking different components in (30). Dual momentum twistors (as well as usual momentum twistors) are reviewed in some detail in appendixA.1, see formula (44) for an explicit expression relating them to the original momentum twistors, includingall factors. Here we are dealing with weight free quantities and as a result, we can always safely drop anynormalization from either Z ’s or W ’s on the left or right hand sides in (30). The latter is symmetric under Z → W so we can evaluate it with either twistors Z or dual twistors W . η -components of the ratio function simply because they do not carry the samehelicity weights. Equating different η -components would be tantamount to comparing applesand oranges. In contrast, when extracting the χ -components as in (34) we generate weightfree quantities since the χ ’s – contrary to the η ’s – carry no weight. This is what allows usto write parity relations at the level of the Wilson loop in terms of simple relations such as(33)–(34) or, simply, in terms of the master relation (30), without the need of dressing thecomponents by additional weight factors.Having decoded in detail the notation behind our main claim (30), let us now explainhow the relations (33)–(34) are nicely suggested by the pentagon approach. Then, we willexplain what sort of checks/derivations we have performed.Parity, first and foremost, is a symmetry that swaps the helicity of the external particlesin the N = 4 supermultiplet that are being scattered, see (29). Similarly, parity also flipsthe helicity of the flux-tube excitations. Flipping the helicity of a flux-tube excitation istrivial: it can be accomplished by simply flipping the signs of all angles φ j ’s, while keepingthe times τ j and distances σ j invariant [1, 2, 4]. This is precisely what the transformation Z ↔ W accomplishes! This explains the substitution rule in the right hand side of (32)–(35). To complete thepicture we also have to act with parity on the pentagon transitions. Naturally, it is expectedto swap the several super pentagon components in (2) in exactly the same way that it actson the usual super-field multiplet expansion (replacing the positive helicity gluon with no˜ η ’s with the negative helicity gluon with 4 ˜ η ’s and so on.). This translates into P ↔ P , P ↔ P etc , (36)which is precisely what is encoded in (33)–(34) or, more generally, in (30). In particular, theseprescriptions neatly relate N k MHV and N n − k − MHV amplitudes, as expected for parity.While (30) is what the OPE naturally suggests, the previous paragraph is obviously nota proof. In any case, (30) is a concrete conjecture for the realization of parity at the Wilsonloop level that we should be able to establish (or disprove) rather straightforwardly startingfrom (29), without any reference whatsoever to the OPE. It would be interesting if a simpleand elegant derivation of (30) existed, perhaps following the same sort of manipulations asin [26]. This would elucidate further the origin of the (weight free) super OPE Grassmannvariables χ .What we did was less thorough. To convince ourselves of the validity of (30) we did twosimpler exercises: On the one hand, using the very convenient package by Bourjaily, Caron-Huot and Trnka [30] we extensively tested (30) for a very large number of ratio functionsfrom NMHV hexagons to N MHV decagons, both at tree and at one loop level. On the More precisely, it is a very instructive exercise to observe that under Z j → W j the cross-ratios in formula(160) in [4] precisely transform as ( τ j , σ j , φ j ) → ( τ j , σ j , − φ j ). When preforming such check it is importantto take into account the conversion between the edge labelling used here and there, see caption of figure 1. When checking such identities for a very large number of edges, the package becomes unpracticallyslow. The trick is to open the package and do a “find/replace operation” to eliminate several
Simplify and
FullSimplify throughout. For analytical checks of relations such as (33)–(34), these simplifications aresuperfluous. P ◦ · · · ◦ P (cid:124) (cid:123)(cid:122) (cid:125) j ◦ P ◦ P (cid:124) (cid:123)(cid:122) (cid:125) n − − j = P ◦ · · · ◦ P (cid:124) (cid:123)(cid:122) (cid:125) j ◦ P ◦ P (cid:124) (cid:123)(cid:122) (cid:125) n − − j | Z → W . (37) In this paper we have constructed a simple map between N k MHV amplitudes and so-called charged pentagon transitions . This map allows one to OPE expand amplitudes with arbitraryhelicity configurations at any value of the coupling.In the dual super-loop description of the amplitude, the charged transitions are operatorsthat act on the color flux tube. They can be realized as combinations of a properly chosensupercharge Q and the more standard bosonic pentagon operator P , and collected into asuper pentagon P = P (cid:89) A =1 (cid:0)
11 + χ A Q A (cid:1) , (38)where the χ ’s are new (weight-free) Grassmann variables associated to the pentagons in thetessellation of the n -gon. The full super loop is then obtained by merging these pentagonstogether, W n = (cid:104) P ◦ P ◦ · · · ◦ P n − (cid:105) , (39)as previously done in (3). The χ -components of the super transition P can be bootstrappedusing the underlying flux-tube integrability, in pretty much the same way as their bosoniccounterparts [4, 6, 12, 13]. In this paper we proposed a map between these χ -components ofthe super Wilson loop W n and its more conventional η -components [10, 11]. This map, givenin (11) and (21), therefore provides the key missing ingredient in the finite coupling OPEexpansion of any helicity amplitude.The map (11) can also be regarded as a definition of the charged transitions. In thisconstruction, we charge a pentagon by acting with the corresponding super-symmetry gen-erators on all the edges at its bottom (top). This is the same as acting on the flux-tube stateentering the pentagon from the bottom (top) – via the flux operator-state correspondence –and is therefore equivalent to (38). This point of view is useful in providing a nice connection between the charged transitionsand their non-charged counterparts. To illustrate this, consider a standard pentagon tran-sition between a fermion and some other state, P ( ¯ ψ A ( p ) | . . . ) = (cid:104) . . . |P| ¯ ψ A ( p ) (cid:105) . As the mo-mentum of this excitation, p , goes to zero, the fermion effectively becomes a super-symmetrygenerator Q A [3, 31]. Therefore, we expect that in this limit this bosonic transition can be This is pretty much the way the super loop was generated in [10, 11]. P A (0 | . . . ) = (cid:104) . . . |P · Q A | (cid:105) . Indeed, while bootstrappingthese transitions we have recently observed curious relations of the sort, P A (0 | . . . ) ∝ (cid:73) p =0 dp π ˆ µ ψ ( p ) P ( ¯ ψ A ( p ) | . . . ) (40)which seems to embody this idea in a rather sharp way.We are currently exploring such directions and their generalizations and will present ourfindings elsewhere. Here we would like to briefly mention two interesting implications:First, in the same way that we considered fermions ¯ ψ with zero momentum, we could alsoconsider adding their conjugate ψ , which effectively becomes the conjugate super-symmetrygenerator ¯ Q . This would naively define a non-chiral super pentagon admitting an expansionboth in χ ’s and in ¯ χ ’s. It is tempting to muse that it should be related to the non-chiralsuper loop proposed in [17] and further studied in [32].Second, the relation (40) between Q and ¯ ψ can be regarded as a local OPE definitionof a charged edge or, equivalently, of the action of ∂/∂η on the super loop. Under suchdefinition, our map (11) translates into a set of relations that includes the SUSY Wardidentities, notably, and that begs to be interpreted directly from the flux tube. Naively, weexpect them to encode certain discontinuities of the OPE series upon edge-crossing of thefermions. It would be fascinating to clarify this point and instructive to see if some simpleOPE contour manipulation could provide a derivation of supersymmetry from the flux tubetheory.Let us end with further outlook. There is by now a very large reservoir of knowledgeon perturbative scattering amplitudes in planar N = 4 SYM theory, tightly related tothe large amount of symmetries they are subject to (originating from both the originaland dual Wilson loop descriptions). On the other hand, we have the pentagon approach,fully non-perturbative and valid all the way from weak to strong coupling. This approachsacrifices some of the most basic symmetries of the amplitudes, such as supersymmetry,parity and cyclicity. In return, it renders the most non-trivial symmetry of all – integrability – both manifest and practical. We think this is a worthy trade off, especially if the moreconventional symmetries can be recovered in the end. Our map (11) is one realization ofthis philosophy, where different amplitudes that are related by supersymmetry are beingassigned to the same OPE series. Moreover, as discussed above, we now start to understandthat supersymmetry and parity also have, after all, a rather natural OPE incarnation. Inour quest for the ultimate solution to the scattering amplitude problem, the next symmetryto attack is probably cyclicity. Hopefully it will also turn out to be easier than we now think!
Acknowledgements
We thank N. Berkovits, J. Bourjaily, S. Caron-Huot, F. Cachazo, J. Maldacena and S. He fornumerous illuminating discussions, most notably concerning parity, and for inspiring remarkson the role of zero-momentum fermion for super amplitudes. We also thank the participantsand organizers of the New Geometric Structures in Scattering Amplitudes program for an19nspiring program and discussions. Research at the Perimeter Institute is supported in partby the Government of Canada through NSERC and by the Province of Ontario throughMRI. The research of A.S. has been supported by the I-CORE Program of the Planningand Budgeting Committee, The Israel Science Foundation (grant No. 1937/12) and theEU-FP7 Marie Curie, CIG fellowship. L.C. and P.V. thank ICTP-SAIFR for warm hospi-tality during the concluding stages of this project. J.C. is funded by the FCT fellowshipSFRH/BD/69084/2010. The research leading to these results has received funding fromthe People Programme (Marie Curie Actions) of the European Union’s Seventh FrameworkProgramme FP7/2007-2013/ under REA Grant Agreement No 317089 (GATIS).
Centro deFisica do Porto is partially funded by the Foundation for Science and Technology of Portugal(FCT).
A More on Geometry, Pentagons and Parity
In this appendix we review some known facts about the geometry of amplitudes and in par-ticular, pentagons. These facts are then used in section A.3 to prove the parity relation (37).
A.1 Variables
Scattering Amplitudes and null polygonal Wilson loops are conventionally parametrized bya plethora of very useful variables. Amongst them, we have momentum twistors Z , spinorhelicity variables λ and their parity conjugate ˜ λ , and dual momentum twistors W . Let usintroduce them in our notation following [27] closely. We shall start by the momentumtwistors Z and construct all other variables from them.A momentum twistor is a four dimensional projective vector Z j ∼ λZ j . It is associatedto each edge of the null polygon, see figure 1. Momentum twistors allow us to parametrizethe shape of the polygon in an unconstrained way, this being one of their main virtues.Moreover, they transform linearly under conformal transformations and are therefore veryuseful when dealing with a conformal theory such as N = 4 SYM.Note the labelling of edges we are using in this paper is tailored from an OPE analysis andis not the conventional cyclic labelling commonly used to describing color ordered partialamplitudes. In particular, in our convention, Z j and Z j +1 (or Z j − ) are not neighbours;instead they nicely face each other in the polygon tessellation, see figure 1. The trivialconversion between our labelling and a more conventional numbering of the edges is presentedin the caption of figure 1.Out of four momentum twistors we can build conformal invariant angle brackets (cid:104) ijkl (cid:105) ≡ (cid:15) abcd Z ai Z bj Z ck Z dl or (cid:104) ijkl (cid:105) ≡ Z i ∧ Z j ∧ Z k ∧ Z l . (41)We construct spinor helicity variables λ by extracting the first two components of each20our dimensional momentum twistors λ i ≡ (cid:18) (cid:19) · Z i . (42)With these spinor helicity variables we can construct Lorentz invariant two dimensional anglebrackets (cid:104) i, j (cid:105) ≡ (cid:15) αβ λ i,α λ j,β or (cid:104) i, j (cid:105) ≡ Z i · I · Z j (43)where I ab is the usual infinite twistor which one can read off from the first definition. Next weintroduce the dual momentum twistors W which can be thought of as the parity conjugateof the Z ’s. The dual momentum twistors are defined by using three neighbouring standardmomentum twistors as W j,a ≡ (cid:15) abcd Z bj − Z cj Z dj +2 (cid:104) j − , j (cid:105)(cid:104) j, j + 2 (cid:105) or W j ≡ Z j − ∧ Z j ∧ Z j +2 (cid:104) j − , j (cid:105)(cid:104) j, j + 2 (cid:105) . (44)Note that with this convenient normalization the dual momentum twistor W j has the oppositehelicity weight as the momentum twistor Z j . For the very bottom and top we need to tweakthe definition (44) due to the non-cyclic labelling we are using. With the dual momentum twistors we can now construct four brackets once more, nowdenoted with square brackets[ ijkl ] ≡ (cid:15) abcd W i,a W j,b W k,c W l,d or [ ijkl ] ≡ W i ∧ W j ∧ W k ∧ W l . (45)Finally, we come to the parity conjugate spinor helicity variables ˜ λ . They can be nowdefined as the last two components of the dual twistors,˜ λ i = (cid:18) (cid:19) · W i . (46)Out of two such twistors we can construct the Lorentz invariant square brackets[ ij ] ≡ (cid:15) ˙ α ˙ β ˜ λ i, ˙ α ˜ λ j, ˙ β or [ ij ] = W i · ˜ I · W j (47) More precisely, we can always apply a global GL (4) rotation U to all the twistors (before extracting thefirst two components) plus a residual GL (2) transformation V to all the spinors (after extracting them fromthe first two components) such that in total λ i ≡ V · (cid:18) (cid:19) · U · Z i . Henceforth we set U and V tobe the identity matrices. Nevertherless, it is worth keeping in mind that sometimes such transformations canbe quite convenient. For instance, the twistors in previous OPE studies – see e.g. appendix of [4] – containseveral zero components and will lead to singular λ ’s if extracted blindly. In those cases, it is quite convenientto preform such generic conformal transformations when constructing the spinor helicity variables. Explicitly, the only tricky definitions are W ≡ Z ∧ Z ∧ Z − (cid:104) , (cid:105)(cid:104) , − (cid:105) , W − ≡ Z ∧ Z − ∧ Z (cid:104) , − (cid:105)(cid:104)− , (cid:105) at the bottom and W n − ≡ Z n − ∧ Z n − ∧ Z n − (cid:104) n − ,n − (cid:105)(cid:104) n − ,n − (cid:105) and W n − ≡ Z n − ∧ Z n − ∧ Z n − (cid:104) n − ,n − (cid:105)(cid:104) n − ,n − (cid:105) at the top, see figure 1. Literally, the transformation (cid:0) λ, ¯ λ (cid:1) → (cid:0) ¯ λ, λ (cid:1) acts on the momentum p µ σ µα ˙ α = λ α ˜ λ ˙ α as a reflection of p since the corresponding Pauli matrix is antisymmetric while all others are symmetric. Once combinedwith an 180 ◦ rotation in the 1-3 plane, we get what is conventionally denoted by parity. In sum, sincerotation symmetries are an obvious symmetry, one often slightly abuses notation to denote as parity anytransformation whose determinant is − I ab can once gain be read off from the first definition.A beautiful outcome of the construction above is that momentum conservation0 = (cid:88) i λ i,α ˜ λ i, ˙ α for α = 1 , α = ˙1 , ˙2 (48)automatically follows from the definitions above. In other words, as is well known, the useof twistors trivializes momentum conservation.To summarize: At this point, each edge of our polygon is endowed with a momentumtwistor Z j , a dual momentum twistor W j and a pair of spinors λ j and ˜ λ j . There are alsoother null segments which play a critical role in our construction: the middle edges thatdefine our tessellation which are represented by the red dashed lines in figure 1 and whosecorresponding momentum twistors are given in the caption of that same figure. We quotehere for convenience: Z middle = (cid:104) j − , j, j + 2 , j − (cid:105) Z j +1 − (cid:104) j − , j, j + 2 , j + 1 (cid:105) Z j − . (49)Let us briefly explain how this equation can be established. This simple exercise beautifullyillustrates the power of Hodges’ momentum twistors when dealing with the geometry of nulllines. First, since Z middle ∧ Z j − , Z middle ∧ Z j +1 and Z j − ∧ Z j +1 all correspond to the sameright cusp in figure 1a, we immediately have that Z middle = αZ j +1 + βZ j − . At the same timethe point Z middle ∧ Z j – where the middle line intercepts the left edge in figure 1a – lies onthe line Z j +2 ∧ Z j + tZ j − ∧ Z j between the two left cusps. As such, the middle twistor is alsoa linear combination of the twistors Z j , Z j − and Z j +2 and thus (cid:104) j, j − , j + 2 , Z middle (cid:105) = 0.This condition immediately fixes the ratio β/α to be as in (49). The normalization of theprojective twistor can be fixed arbitrarily with (49) being one such choice. Following thelogic above, we can now also associate to each middle edge a dual twistor W middle and a pairof spinors λ middle and ˜ λ middle . They will indeed show up below.We close this section with two useful identities which we shall use latter. The first is (cid:104) i ˆ ij ˆ j (cid:105) [ i ˆ ij ˆ j ] = (cid:104) i ˆ i (cid:105)(cid:104) j ˆ j (cid:105) [ i ˆ i ][ j ˆ j ] (50)where ˆ i and i are neighbouring edges and so are ˆ j and j . The second is (cid:104) abcd (cid:105) = (cid:104) ab (cid:105)(cid:104) bc (cid:105)(cid:104) cd (cid:105) [ bc ] and [ abcd ] = [ ab ][ bc ][ cd ] (cid:104) bc (cid:105) (51)which holds for any four consecutive twistors (starting with a followed by b , then c and then d at the end). Note that the second equality in (51) follows from the first equality theretogether with (50). It also follows trivially from the first equality in (51) under parity whichsimply interchanges square and angle brackets. A.2 Pentagons and Weights
In a tessellation of an n -sided polygon, each two consecutive null squares form a pentagon.As depicted in figure 1, each such pentagon shares some edges with the larger polygon whilesome (either one or two) edges are middle edges defined by the tessellation, see also (49).22hese pentagons play a prominent role in our construction. In particular, here we wantto describe their importance in defining the weight of a given edge with respect to a givenpentagon . To simplify our discussion we label the edges of a generic pentagon as a, b, c, d, e . Pentagons have no cross-ratios. Nevertheless, they are not totally trivial. For instance,they allow us to read of the weight of an edge of the pentagon (with respect to that pentagon)through the pentagon NMHV ratio function components as R ( abcd ) = 1 abcd , R ( aabc ) = 1 a bc , R ( aaaa ) = 1 a , (52)and so on. All such components can be extracted from a single R-invariant beautifullywritten using momentum twistors in [26, 27], R NMHV pentagon = (cid:81) A =1 ( (cid:104) abcd (cid:105) η Ae + (cid:104) bcde (cid:105) η Aa + (cid:104) cdea (cid:105) η Ab + (cid:104) deab (cid:105) η Ac + (cid:104) eabc (cid:105) η Ad ) (cid:104) abcd (cid:105)(cid:104) bcde (cid:105)(cid:104) cdea (cid:105)(cid:104) deab (cid:105)(cid:104) eabc (cid:105) . (53)From the relations (52) we read a = (cid:104) abcd (cid:105)(cid:104) cdea (cid:105)(cid:104) deab (cid:105)(cid:104) eabc (cid:105)(cid:104) bcde (cid:105) . (54)We can also re-write this relation using (51) as a = (cid:104) ab (cid:105) (cid:104) ea (cid:105) (cid:104) ab (cid:105)(cid:104) bc (cid:105)(cid:104) de (cid:105)(cid:104) ea (cid:105)(cid:104) cd (cid:105) / [ cd ] [ ab ][ bc ][ cd ][ de ][ ea ] (55)where the familiar Parke-Taylor chains nicely show up.Furthermore, note that a product of three weights with respect to the same pentagoncan be traded by the weight of any of the other two twistors of the pentagon using the firstrelation in (52) with R ( abcd ) = 1 / (cid:104) abcd (cid:105) . In particular, it follows that1 bce = a (cid:104) abce (cid:105) = d (cid:104) dbce (cid:105) . (56)This allows us to massage slightly some of the formulae in the main text. For example, (11)can be written a bit more economically using1( j − ) j ( j ) j ( j + ) j = ( t j ) j W j · Z t j or 1( j − ) j ( j ) j ( j + ) j = ( b j ) j W j · Z b j (57)where t j and b j indicate the top or bottom twistors of pentagon j respectively, see figure 7.Note that it is irrelevant that we do not fix the normalizations of these top and bottomtwistors: they drop out in the ratios here constructed. For example, this pentagon could be the first pentagon in the tessellation in figure 1b. In this case wewould set a = Z , b = Z , c = Z − , d = Z and e = Z middle line ending on edge 2 . j − j +1 j +2 j +3 j − j − t j b j Figure 7: The weight factor ( j − ) j ( j ) j ( j + ) j associated to the j -th pentagon can be expressedin terms of the twistors of the larger polygon. It involves the seven closest edges to that pentagon,as illustrated in the figure. It is clear from this example the advantage of using this edge labellingas opposed to the cyclic one. We can also explicitly evaluate (57) by plugging in (54) the expressions for the middlemomentum twistors (49), see figure 7. When doing so, one finds (cid:18) j − ) j ( j ) j ( j + ) j (cid:19) = (58)= (cid:104) j − , j − , j + 1 , j + 3 (cid:105)(cid:104) j − , j − , j, j + 2 (cid:105)(cid:104) j − , j, j + 1 , j + 2 (cid:105)(cid:104) j − , j − , j, j + 1 (cid:105) (cid:104) j − , j, j + 1 , j + 2 (cid:105) (cid:104) j − , j, j + 1 , j + 3 (cid:105)(cid:104) j − , j − , j, j + 1 (cid:105) . A.3 Parity Map
In this section we establish the parity relation (37) which we can equivalently cast as (cid:18) ∂∂χ (cid:19) . . . (cid:18) ∂∂χ j (cid:19) R = (cid:18) ∂∂χ j +1 (cid:19) . . . (cid:18) ∂∂χ n − (cid:19) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z → W . (59)To evaluate the left hand side we note that ∂∂χ k = (cid:104) k − , k, k + 1 , k − (cid:105) ( k − ) k ( k ) j ( k + ) k ∂∂η k − + . . . (60)where the . . . contain a linear combination of derivatives from ∂/∂η − until ∂/∂η k − . Sincewe are taking the maximum number of each derivative ∂/∂χ k , only the term written in (60)contributes, while all other terms are already saturated by the previous derivatives. There-fore, at the end we are left with a single ratio function component R ( − ... ( j − ≡ R ( − , − , − , − ,..., ( j − ,j − ,j − ,j − . As usual, for the bottom and top pentagons we need to adjust this formula slightly. For instance, for j = 1 we find Z − in the right hand side which is not defined, see figure 1. The fix is very simple: we shouldsimply replace Z − by the very bottom twistor, that is Z − . Similarly for the top pentagon, where we shouldreplace Z n − by the very top twistor Z n − .
24e can proceed in a similar fashion for the right hand side of (59), restricting the sum in(11) to the edges above that pentagon. Keeping track of the multiplicative weight factors,we can now rewrite (59) as R ( − ... ( j − R ( n − ... ( j +3) (cid:12)(cid:12) Z → W = (cid:32) n (cid:89) k = j +1 (cid:104) k − , k, k + 1 , k + 2 (cid:105) ( k − ) k ( k ) k ( k + ) k (cid:33) Z → W / (cid:32) j (cid:89) k =1 (cid:104) k − , k, k + 1 , k − (cid:105) ( k − ) k ( k ) k ( k + ) k (cid:33) . (61)At this point, it is convenient to revert back to a more conventional cyclic notation. Weshall revert to cyclic variables using the map in caption of figure 1 followed by a simple overallcyclic rotation of all the indices (by a convenient n and j dependent amount). Altogether,we map each edge index k in (61) as k → n − j − δ j odd − k δ k even + k + 32 δ k odd . (62)This change of labelling is illustrated in figure 8 for n = 8 and j = 3. To avoid anyconfusions, we will add a C to the label of all equations written in this cyclic labelling. Next,it is useful to convert the ratio in (61) to two-brackets using (55), (50) and (51). In two-bracket notation, the parity transformation Z → W simply amounts to interchanging squareand angle brackets. At the end of the day, we arrive at the nice expression R ( n ) ... ( n − j +1) R ( n − j − ... (3) | Z → W = (cid:104) , (cid:105) . . . (cid:104) n, (cid:105)(cid:104) n, (cid:105) . . . (cid:104) n − j, n − j + 1 (cid:105) [ n − j − , n − j − . . . [2 , [1 , . . . [ n, . ( C λ ↔ ˜ λ and the usual Grassmann variables ˜ η ↔ ¯˜ η , see (29). Weshall show that ( C
63) is a simple consequence of the more transparent relation A n [1 − , + , ... , ( n − j − + , ( n − j ) − , ... , n − ] = A n [1 + , − , ... , ( n − j − − , ( n − j ) + , ... , n + ] ∗ ( C j + 2 negative helicity and n − j − η ) (˜ η n − j ) . . . (˜ η n ) of the super amplitude A n = δ ( n (cid:80) i =1 λ i ˜ η i ) (cid:104) , (cid:105) . . . (cid:104) n, (cid:105) M MHV loop n ( λ, ˜ λ ) R ( η, Z ) , ( C R = 1 + R NMHV + . . . is the ratio function and M MHV loop n ( λ, ˜ λ ) is the MHV amplitudedivided by its tree level part. To extract this component we need to recall the relation When simplifying the ratio of weights it is convenient to explore momentum conservation for the variousmiddle squares to see that the dependence on the middle spinors neatly drops out. Recall that for any squaremomentum conservation (cid:80) j =1 λ j ˜ λ j = 0 readily leads to (cid:104) , (cid:105) [2 ,
3] = −(cid:104) , (cid:105) [4 ,
3] and (cid:104) , (cid:105) [2 ,
1] = (cid:104) , (cid:105) [3 , Figure 8: Example for N MHV octagon, ( n = 8, j = 3). In blue, the edges charged for P ◦P ◦ P ◦ P and in green the edge charged for the parity conjugate P ◦ P ◦ P ◦ P . The OPElabelling for the edges is presented in black and in red the cyclic labelling used in this derivation. between the Grassmann variables ˜ η and η showing up in this expression. In one direction, itreads [26] ˜ η i = (cid:104) i, i + 1 (cid:105) η i − + (cid:104) i + 1 , i − (cid:105) η i + (cid:104) i − , i (cid:105) η i +1 (cid:104) i − , i (cid:105)(cid:104) i, i + 1 (cid:105) , ( C η = η = 0 and [28, 29] η = (cid:104) , (cid:105) ˜ η ,η = (cid:104) , (cid:105) ˜ η + (cid:104) , (cid:105) ˜ η , ... η n = (cid:104) , n (cid:105) ˜ η + . . . + (cid:104) n − , n (cid:105) ˜ η n − . ( C η and ˜ η n do not appear in this inverse map, we must look for them in the fermionic deltafunction when extracting this component. Therefore, we can simply replace the fermionicdelta function by (cid:104) n, (cid:105) and consider the component (˜ η n − j ) . . . (˜ η n − ) of the simpler quan-tity (cid:104) n, (cid:105) (cid:104) , (cid:105) . . . (cid:104) n, (cid:105) M MHV loop n ( λ, ˜ λ ) R ( η, Z ) (cid:12)(cid:12)(cid:12)(cid:12) η j = (cid:80) j − k =2 (cid:104) k,j (cid:105) ˜ η k . ( C η n − shows up only in η n such that extracting four units ofit is tantamount to taking four powers of η n (times (cid:104) n − , n (cid:105) ). Next, η n is crossed out and26 η n − shows up in η n − only and so on. All in all, we arrive at A n [1 − , + , ... , ( n − j − + , ( n − j ) − , ... , n − ] = ( C M MHV loop n ( λ, ˜ λ ) (cid:104) n, (cid:105) (cid:104) , (cid:105) . . . (cid:104) n, (cid:105) (cid:104) n − , n (cid:105) . . . (cid:104) n − j, n − j + 1 (cid:105) R ( n ) ... ( n − j +1) . The right hand side of ( C
64) can be treated similarly. In the end, we conclude that (cid:104) n, (cid:105) (cid:104) , (cid:105) . . . (cid:104) n, (cid:105) (cid:104) n − , n (cid:105) . . . (cid:104) n − j, n − j + 1 (cid:105) R ( n ) ... ( n − j +1) ( Z )= [ n − j − , n − j − [1 , . . . [ n,
1] [ n − j − , n − j − . . . [2 , R ( n − j − ... (3) ( W ) ( C C
63) thus proving (37). This was the main goal of thisappendix.Other cases can be analyzed in a similar way. For instance, to establish an identity like P ◦ . . . ◦ P ◦ P ◦ P ◦ P ◦ . . . ◦ P = P ◦ . . . ◦ P ◦ P ◦ P ◦ P ◦ . . . ◦ P | Z → W westart – on the amplitude side – with an amplitude that besides gluons also involves a positivehelicity fermion ψ and one negative helicity fermion ¯ ψ . However, the analysis becomes moreand more cumbersome as we consider cases with pentagons that are further away from beingmaximally charged. It would be interesting to streamline this analysis and work out thegeneral case in a clean way. A better understanding of (the space of) all possible inversemaps η (˜ η ) would probably be useful in this respect. References [1] B. Basso, A. Sever and P. Vieira, “Spacetime and Flux Tube S-Matrices at Finite Cou-pling for N=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. (2013) 9,091602 [arXiv:1303.1396 [hep-th]].[2] L. F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, “An Operator ProductExpansion for Polygonal null Wilson Loops,” JHEP (2011) 088 [arXiv:1006.2788[hep-th]]. • A. Sever and P. Vieira, “Multichannel Conformal Blocks for Polygon WilsonLoops,” JHEP (2012) 070 [arXiv:1105.5748 [hep-th]].[3] B. Basso, “Exciting the GKP string at any coupling,” Nucl. Phys. B (2012) 254[arXiv:1010.5237].[4] B. Basso, A. Sever and P. Vieira, “Space-time S-matrix and Flux tube S-matrix II.Extracting and Matching Data,” JHEP (2014) 008 [arXiv:1306.2058 [hep-th]]. When doing so it is convenient to note that the right hand side of ( C
64) can be also written as( A n [1 − , + , ... , ( j +3) + , ( j +4) − , ... , n − ]) ∗ i → i − j − . In this form, it is clear that we can simply recycle the result( C
69) with the obvious replacement of angle brackets by square brackets following from the conjugation. (2014) 149 [arXiv:1407.1736 [hep-th]].[7] B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, To Appear[8] L. F. Alday, J. M. Maldacena, “Gluon scattering amplitudes at strong coupling,” JHEP , (2007) 064 [arXiv:0705.0303].[9] G. P. Korchemsky, J. M. Drummond, E. Sokatchev, “Conformal properties of four-gluon planar amplitudes and Wilson loops,” Nucl. Phys.
B795 , (2008) 385-408[arXiv:0707.0243] • A. Brandhuber, P. Heslop, G. Travaglini, “MHV amplitudesin N=4 super Yang-Mills and Wilson loops,” Nucl. Phys.
B794 , (2008) 231-243[arXiv:0707.1153] • Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin,C. Vergu and A. Volovich, “The Two-Loop Six-Gluon MHV Amplitude in MaximallySupersymmetric Yang-Mills Theory,” Phys. Rev. D , (2008) 045007 [arXiv:0803.1465] • J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “Hexagon Wilsonloop = six-gluon MHV amplitude,” Nucl. Phys. B (2009) 142 [arXiv:0803.1466] • N. Berkovits, J. Maldacena, “Fermionic T-Duality, Dual Superconformal Symmetry, andthe Amplitude/Wilson Loop Connection,” JHEP , (2008) 062 [arXiv:0807.3196].[10] L. J. Mason, D. Skinner, “The Complete Planar S-matrix of N=4 SYM as a WilsonLoop in Twistor Space,” JHEP , (2010) 018 [arXiv:1009.2225].[11] S. Caron-Huot, “Notes on the scattering amplitude / Wilson loop duality,”[arXiv:1010.1167].[12] A. V. Belitsky, “A note on two-loop superloop,” Phys. Lett. B (2012) 205[arXiv:1207.1924 [hep-th]].[13] A. V. Belitsky, “Nonsinglet pentagons and NHMV amplitudes,” arXiv:1407.2853 [hep-th].[14] H. Elvang, D. Z. Freedman and M. Kiermaier, “Solution to the Ward Identities forSuperamplitudes,” JHEP (2010) 103 [arXiv:0911.3169 [hep-th]].[15] V. P. Nair, “A Current Algebra For Some Gauge Theory Amplitudes,” Phys. Lett. B , 215 (1988).[16] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “Dual superconformalsymmetry of scattering amplitudes in N=4 super-Yang-Mills theory,” Nucl. Phys. B (2010) 317 [arXiv:0807.1095].[17] S. Caron-Huot, “Superconformal symmetry and two-loop amplitudes in planar N=4super Yang-Mills,” JHEP (2011) 066 [arXiv:1105.5606].[18] G. P. Korchemsky and E. Sokatchev, “Superconformal invariants for scattering ampli-tudes in N=4 SYM theory,” Nucl. Phys. B (2010) 377 [arXiv:1002.4625 [hep-th]].2819] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, “The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM,” JHEP (2011) 041[arXiv:1008.2958 [hep-th]]. • N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Gon-charov, A. Postnikov and J. Trnka, “Scattering Amplitudes and the Positive Grassman-nian,” arXiv:1212.5605 [hep-th]. • N. Arkani-Hamed and J. Trnka, “The Amplituhe-dron,” arXiv:1312.2007 [hep-th].[20] A. V. Belitsky, “Fermionic pentagons and NMHV hexagon,” arXiv:1410.2534 [hep-th].[21] A. Sever, P. Vieira and T. Wang, “OPE for Super Loops,” JHEP (2011) 051[arXiv:1108.1575 [hep-th]].[22] A. Sever, P. Vieira and T. Wang, “From Polygon Wilson Loops to Spin Chains andBack,” JHEP (2012) 065 [arXiv:1208.0841 [hep-th]].[23] Tree Heptagon exercise on the first day of the 6th edition of the Mathematica Schoolhttp://msstp.org/?q=node/289.[24] L. J. Dixon, J. M. Drummond and J. M. Henn, “Bootstrapping the three-loop hexagon,”JHEP (2011) 023 [arXiv:1108.4461]. • L. J. Dixon, J. M. Drummond, C. Duhr andJ. Pennington, “The four-loop remainder function and multi-Regge behavior at NNLLAin planar N = 4 super-Yang-Mills theory,” JHEP (2014) 116 [arXiv:1402.3300 [hep-th]]. • L. J. Dixon, J. M. Drummond, C. Duhr, M. von Hippel and J. Pennington, “Boot-strapping six-gluon scattering in planar N=4 super-Yang-Mills theory,” PoS LL (2014) 077 [arXiv:1407.4724 [hep-th]]. • L. J. Dixon and M. von Hippel, “Bootstrappingan NMHV amplitude through three loops,” JHEP (2014) 65 [arXiv:1408.1505[hep-th]].[25] N. Arkani-Hamed, F. Cachazo and J. Kaplan, “What is the Simplest Quantum FieldTheory?,” JHEP (2010) 016 [arXiv:0808.1446 [hep-th]].[26] L. J. Mason and D. Skinner, “Dual Superconformal Invariance, Momentum Twistorsand Grassmannians,” JHEP (2009) 045 [arXiv:0909.0250 [hep-th]].[27] A. Hodges, “Eliminating spurious poles from gauge-theoretic amplitudes,” JHEP (2013) 135 [arXiv:0905.1473 [hep-th]].[28] J. L. Bourjaily, “Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion inMathematica,” arXiv:1011.2447 [hep-ph].[29] G. P. Korchemsky and E. Sokatchev, “Symmetries and analytic properties of scatteringamplitudes in N=4 SYM theory,” Nucl. Phys. B , 1 (2010) [arXiv:0906.1737 [hep-th]].[30] J. L. Bourjaily, S. Caron-Huot and J. Trnka, “Dual-Conformal Regularization of InfraredLoop Divergences and the Chiral Box Expansion,” arXiv:1303.4734 [hep-th].[31] L. F. Alday and J. M. Maldacena, “Comments on operators with large spin,” JHEP (2007) 019 [arXiv:0708.0672]. 2932] N. Beisert, S. He, B. U. W. Schwab and C. Vergu, “Null Polygonal Wilson Loops in FullN=4 Superspace,” J. Phys. A45