Dual Conformal Structure Beyond the Planar Limit

The planar scattering amplitudes of $\mathcal{N} = 4$ super-Yang--Mills theory display symmetries and structures which underlie their relatively simple analytic properties such as having only logarithmic singularities and no poles at infinity. Recent work shows in various nontrivial examples that the simple analytic properties of the planar sector survive into the nonplanar sector, but this has yet to be understood from underlying symmetries. Here we explicitly show that for an infinite class of nonplanar integrals that covers all subleading-color contributions to the two-loop four- and five-point amplitudes of $\mathcal{N} = 4$ super-Yang--Mills theory, symmetries analogous to dual conformal invariance exist. A natural conjecture is that this continues to all amplitudes of the theory at any loop order.

Introduction. Recent years have seen significant advances in constructing scattering amplitudes, especially for planar N = 4 super-Yang-Mills (sYM) theory. A key feature of planar N = 4 sYM theory that makes this progress possible is its remarkable symmetries and structures. These include dual conformal symmetry [1], Yangian symmetry [2], integrability [3], a dual interpretation of scattering amplitudes in terms of Wilson loops [4], uniform transcendentality [5], structures that aid various bootstraps [6,7], and even an all-loop resummation of four-and five-point amplitudes [8]. Scattering amplitudes have been reformulated using on-shell diagrams and the positive Grassmannian [9], which culminated in the geometric concept of the amplituhedron [10]. Some of these advances have been helpful in quantum chromodynamics relevant for collider physics, including improved ways for dealing with polylogarithms that arise in multiloop computations [11] and for finding good choices [12][13][14] of integral bases that simplify their evaluation. In fact, the integrals we analyze here for the two-loop five-point amplitude [14,15] are useful choices for the basis of master integrals for 2-to-3 scattering in generic theories [16].
These symmetries and structures impose nontrivial constraints on the analytic properties of planar N = 4 sYM amplitudes. In particular, the loop-level colorordered amplitudes M 123...n can be written as where the integrand I has only logarithmic singularities, no poles at infinity [9], and unit leading singularities [17] as tied to the amplituhedron [10]. The prefactor PT 123...n is the standard Parke-Taylor factor [18], as defined in e.g. Ref. [14]. It is unclear how to define dual conformal symmetry in the nonplanar sector given the lack of dual variables to define the symmetry. However, as shown in a variety of examples [13,14,19], the key analytic properties of the planar sector implied by its symmetries carry over to the nonplanar sector, even if the symmetries are unclear. In each example, the full amplitude can be expressed as [20] where the a σ,k,j are rational numbers, the c k are color factors, the PT σ are the Parke-Taylor factors corresponding to an ordering σ of external particles, and the I j are integrands with only logarithmic singularities, no poles at infinity, and unit leading singularities. Eq. (2) is a natural extension of Eq. (1) to the nonplanar sector. It is nontrivial that such a representation exists where each integrand is expressed in terms of local diagrams. Some structures of the non-planar sector were also explored at the level of on-shell diagrams [21,22].
In the present paper we address the following question: Can we identify a hidden symmetry associated with the simple analytic properties for the nonplanar sector uncovered in Refs. [13,14,19]? Building on the initial studies in Ref. [23], we answer this question affirmatively and demonstrate that the integrands I j in (2) encoding the simple analytic structure of the full two-loop fourand five-point amplitudes all have hidden symmetries related to dual conformal invariance. These are not hidden symmetries of the full amplitude, but of individual components of the amplitudes, analogous to the situation with dual conformal symmetry in the planar case (1). We also identify an infinite class of nonplanar integrands with the hidden symmetry. In many cases these symmetries rely on nontrivial identities, making it all the more striking that a symmetry actually exists.
Dual coordinates and conformal symmetry. To set up our discussion of hidden symmetries in the nonplanar sector, we first briefly review dual conformal symmetry in the planar sector [1]. In general, the momenta (corresponding to edges or lines) in any planar diagram can be represented as the difference of adjacent dual coordinates (corresponding to regions). For example, the momenta in the planar double-box diagram on the left of Figure 1 can be expressed as where the p i are external momenta, l 5 and l 6 are the loop momenta, and x i are the dual coordinates with all Lorentz indices omitted. We can perform infinitesimal conformal transformations on these dual coordinates, where b µ is an infinitesimal boost vector. The transformation of the square of proper distance is In general, if a quantity f transforms as δf = wf with w a local function, we say f rescales under the transformation with weight w. Thus, under dual conformal transformations, (x i −x j ) 2 carries a weight −b·(x i +x j ). Note that all massless external legs remain on-shell after the transformation. All the inverse propagators have the form (x i − x j ) 2 . This implies that locality is maintained for planar loop integrals under dual conformal transformations and allows us to construct simple functions that are invariant. As a simple illustration, consider an integral associated with the planar double box, The inverse Feynman propagators ρ k in dual coordinates are In what follows, we will be interested in the integrand I, defined by I =´I. With this numerator the integrand has a hidden symmetry exposed by using the dual variables [1]. Performing the dual conformal transformation on the integrand (6) yields where we used For D = 4 space-time dimensions this integrand is invariant under dual conformal transformations, which is what motivated the choice of numerator s 2 t. Outside D = 4, this is reminiscent of -form differential equations [12], but without doubled propagators on the right hand side before reduction to a basis [23,24]. What is the relevance of this symmetry? It turns out that all integrands of planar N = 4 sYM amplitudes possess this property, which then leads to nontrivial constraints on the amplitude after integration. This is the celebrated dual conformal symmetry [1] which has spurred many developments. In the following we identify an analogous symmetry in a class of nonplanar diagrams.
Nonplanar extension. While there are no known global variables for generic nonplanar diagrams, it is natural to require that, as for the planar case, a non-planar analog of dual conformal transformations also maintains the local structure for inverse propagators, δρ k ∝ ρ k [23]. We start by considering a nonplanar diagram that can be made planar by moving the location of one external leg carrying momentum p µ k . This is an infinite class of nonplanar integrals, and includes all the nonplanar integrals at two loops with five or less external legs. In particular, all of the nonplanar integrals in Figure 2 are of this type. For example diagram (a) can be made planar by moving external leg 3. Under this, the momenta of the propagators are modified compared to the planar case at most by adding or subtracting a single external momentum p µ k . Thus, the inverse propagators ρ l therein can be written as either (x i − x j ) 2 , or (x i − x j ± p k ) 2 , when using the dual coordinates of the planar cousin. The key observation here is that if the infinitesimal boost vector b µ is proportional to a massless external leg p µ k , then (x i − x j ± p k ) 2 transforms in the same way as (x i − x j ) 2 for any x µ i and x µ j . Specifically, implying that all the propagators in this class of nonplanar diagrams satisfy δρ k ∝ ρ k for this conformal boost. As a simple first example, consider the crossed doublebox diagram on the right of Fig. 1, with numerator N 1 = su(l 5 + p 4 ) 2 , which is one of the nonplanar pure integrands found in Ref. [19] as a building block of the full amplitude: where the ρ k are the inverse propagators. This diagram can be obtained from the planar double box in Fig. 1 by moving the external leg 3 to the central rung. Using the dual coordinates of the planar double box, we can write the nonplanar integrand as where the propagators are given by with the x i defined in Eq. (3). Applying a dual conformal transformation to the integrand with the boost vector b µ ∝ p µ 3 and using equation (10) we find that exposing a hidden symmetry in D = 4. A similar analysis holds for the numerator N 2 = st(l 5 + p 3 ) 2 , corresponding to the other pure integrand found in Ref. [19]. One can also obtain the crossed box from the planar double box by moving the leg 4 to the central rung (and making a change in the momentum routing of the planar double box), giving a new conformal boost with b µ ∝ p µ 4 . As can be straightforwardly checked, both numerators N 1 and N 2 give integrands that are invariant in D = 4 under this transformation as well.
While we propose these transformations as a natural extension of the planar dual conformal symmetry, it is striking that the numerators N 1 and N 2 are precisely the correct numerators of the building blocks for the two-loop four-point amplitude in N = 4 sYM that unveil their analytic properties [19]. Here we see that we can constrain these numerators from symmetry considerations instead of from imposing desired analytic properties on the integrands. Similar symmetry considerations can be used to match the numerators of a subset of three-loop four-point diagrams in Ref. [13] that can be obtained from planar ones by moving a single external line.
Two-loop five-point case. As the central nontrivial example consider the two-loop five-point N = 4 sYM amplitude first obtained in Ref. [15]. This amplitude was rewritten in a desired form where each diagram composing the amplitude contains only logarithmic singularities and no pole at infinity [14], as follows from dual conformal symmetry in the planar case. The diagrams composing this amplitude are given in Fig. 2. These diagrams are either planar, or in the nonplanar class of diagrams discussed above, so our discussion immediately applies.
Consider diagrams (a), (d), (h), and (i), which can be made planar by moving the external leg 3, corresponding to choosing b µ ∝ p µ 3 . Using the dual coordinates  in the diagram on the right of Figure 3, the propagators in the original nonplanar diagrams are a subset of A crucial difference between integrands at four points and five points is the appearance of spinor helicity variables, which makes the transformation properties less clear. We therefore restrict to D = 4 from now on, and the convention for spinors is chosen such that A complete set of numerators for the diagrams in Figure 2 is given in Table 3 of Ref. [14].
To warm up, consider the numerator in diagram (i) This numerator is constructed to follow the S 3 symmetry among legs 2, 3, 5 of the diagram (up to a sign). By choosing to move leg 3 to make the diagram planar and using the coordinates in Eq. (15), we recast the numerator as under momentum conservation and spinor identities. To see that this numerator only rescales with a local weight under the transformation with b µ ∝ p µ 3 , we need a nontrivial identity . We have confirmed Eq. (19) numerically through n = 8, irrespective of whether the x ij 's are null separated or not. Therefore the numerator in Eq. (18) is manifestly rescaled under the transformation with weight −b · (x 2 + x 3 + x 4 + x 5 ). Moreover, accounting for the transformation of the propagators and measure using Eqs. (9), (10), and (16), this is precisely the weight needed to make the integrand invariant.
We can also make diagram (i) planar by moving the leg carrying momentum p 2 or p 5 , giving a total of three choices of b µ for the conformal boosts. We have checked that these three transformations are independent symmetry generators, corresponding to three hidden symmetries of this nonplanar integrand.
A more involved example is diagram (a) in Fig. 2. The numerator yielding the desired analytic properties given in Ref. [14] is where l * 7 = [54] [24] |5 [2| and l * 6 = p 1 + [23] [13] |2 [1|. How this numerator transforms is far from clear in the above form. In fact, the first or second term alone does not rescale with a local weight. However, by using on-shell conditions and Schouten identities it can be rewritten as using the dual coordinates in Eq. (15). With the help of equation (19), each of the two terms in Eq. (21) above transform with the weight necessary to make the integrand invariant in D = 4. After canceling the propagators, the first term gives rise to the daughter diagram (i) in Fig. 2, and the numerator 3|x 23 x 34 x 45 |3] also matches to one of the components in Eq. (18).
Similarly, we can rewrite the original numerators of diagrams (d) and (h) using the dual coordinates in the diagram on the left of  In addition there are numerators simply related via diagram symmetries. Using Eqs. (9), (10), and (19), we see that these numerators have weights that make the integrand invariant under the dual conformal boost with b µ ∝ p µ 3 . Diagrams (c) and (f) can be made planar by moving the external leg carrying momentum p 4 , corresponding to b µ ∝ p µ 4 . The dual coordinates are defined according to the left of Figure 3, analogous to Eq. (15). The propagators in the original nonplanar diagrams are a subset of The numerator of diagram (f) is N with the same l * 6 as defined below Eq. (20). After canceling the propagator, the first term matches N (f) 2 of Ref. [14] which is related to N (f) 1 under 4 ↔ 5.
We have checked all of the two-loop five-point nonplanar integrands from Ref. [14] that manifest the desired analytic properties of the full two-loop five-point amplitude and found that all of them have a hidden symmetry in D = 4 closely related to dual conformal symmetry. In cases where more than one conformal boost is available, as for diagrams (c), (f), (h), and (i) in Figure 2, we have checked that all such choices of b µ give symmetries of the integrand. While Eq. (10) guarantees that all the propagators transform with definite weight, the fact that all the corresponding numerators behave accordingly to make the integrand invariant appears miraculous.
Using equations (10) and (19) we can generalize these results to integrals relevant for higher-point amplitudes. As a concrete example, consider diagram (a) in Figure 2 but with legs 1,2,4,5 being massive or replaced with arbitrary collections of massless particles, while keeping leg 3 massless. Crucially, the identity in Eq. (19) holds even for x 2 i,i+1 = 0. This implies the numerator with the dual variables in Eq. (15) 3|x 23 x 34 x 45 x 57 x 76 x 61 transforms with the proper weight to make the integrand invariant, providing a generalization of the second term in Eq. (21). Another possible numerator is The latter example (27) is especially interesting since it vanishes in the collinear limit x µ 76 ∝ p µ 3 and gives an infrared-finite integral, for which the hidden symmetry is exact and free of anomalies from divergences. By working in six dimensions, additional finite integrals with the hidden symmetry can be found; such integrals are related to four-dimensional ones via dimension shifting relations [25].
Conclusions. Following the four-point hints in Ref. [23], here we demonstrated that all sectors of the two-loop five-point N = 4 sYM amplitude, including the nonplanar sector, possess new nontrivial hidden symmetries related to dual conformal symmetry. To show this we demonstrated that each integrand sector identified in Ref. [14] possessing simple analytic properties manifests a hidden symmetry. For some sectors the symmetry is rather unobvious. The construction used for the twoloop five-point amplitude extends to any number of loops and legs, giving an infinite class of integrands with new hidden symmetries. It would be interesting to check if these cases actually appear with nonzero coefficient in N = 4 sYM amplitudes. Even for the cases studied here we can expect a larger set of symmetries than the ones we found; we expect this to be helpful for the important problem of identifying the hidden symmetries of more general cases beyond the ones studied here. It would be interesting to apply the symmetries to help identify nonplanar integrals of uniform transcendentality, which become nontrivial at high loop orders by directly checking leading singularities [26]. It would also be interesting to understand how the new symmetries described here relate to recent progress in extending integrability to nonplanar theories described in Ref. [27]. Given the useful role hidden symmetries have played in the planar sector of N = 4 sYM theory, we should expect new progress from fully unraveling the corresponding symmetries of the non-planar sector of the theory.