The Double Pentaladder Integral to All Orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.


Introduction
Despite substantial progress, our understanding of particle scattering in perturbative quantum field theory remains incomplete. One might think that this is to be expected, and that perturbation theory inherently limits us to order-by-order progress in the number of loops. However, the last decade has seen the development of powerful new methods to address scattering at the multi-loop level in both gauge and gravity theories -see, for example, refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and refs. [12,[15][16][17][18][19]. Many of these methods are expected to function up to any desired order in perturbation theory. In the case of planar N = 4 super-Yang Mills (SYM) theory [20,21] there is even an all-orders geometric formulation [7]. Rather, our understanding is incomplete because most of these methods supply, not scattering amplitudes, but integrands depending on loop momenta. Evaluating the multi-loop Feynman integrals produced by these methods is a substantial endeavor in its own right, with two loops only just beginning to yield to systematic analysis [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. In general, perturbative scattering amplitudes are complicated transcendental functions of momentum invariants. If we want to understand these amplitudes to all orders, then we need to understand how to compute these functions order by order, and further, how to sum them into all-orders expressions.
It is not obvious that this is possible in general. However, in the planar limit of N = 4 SYM we have unique evidence that it should be, due to the presence of integrability [41]. Integrability has been used to compute the theory's cusp anomalous dimension for finite coupling [42] and has been instrumental in the Pentagon Operator Product Expansion, which calculates finite-coupling amplitudes in an expansion around a kinematic limit [43][44][45][46][47][48][49]. Notably, the perturbative expansion of these formulas has a finite radius of convergence in the coupling. The kinematic dependence of four-and five-particle amplitudes in planar N = 4 SYM is also captured to all loop orders by the BDS ansatz [50], which is uniquely dictated by the theory's dual conformal symmetry [1,[51][52][53][54][55]. While this symmetry does not uniquely fix the form of amplitudes involving more than five particles, it does restrict the problem to a special class of dual conformally invariant (DCI) integrals [51,[56][57][58], and by extension restricts the form and kinematic dependence of these amplitudes at finite coupling.
As a consequence, a great deal is known about the space of functions that can contribute to six-and seven-particle perturbative amplitudes in planar N = 4 SYM. When these amplitudes are normalized by the BDS ansatz, they can be written in terms of dual superconformal invariants (that encode the helicity structure) multiplied by multiple polylogarithms that have known kinematic dependence and branch cuts only in physical channels [59][60][61][62][63]. Additional physical constraints come from the Steinmann relations, which imply that double discontinuities of amplitudes must be zero when taken in overlapping channels [64][65][66]. These relations are obeyed by the polylogarithmic part of the amplitude when it is normalized by the BDS-like ansatz (which contains only two-particle kinematic invariants) [67][68][69]. We refer to this space of multiple polylogarithms as the space of Steinmann hexagon functions (H) and Steinmann heptagon functions for six-and seven-point kinematics, respectively. These function spaces have proven sufficient to describe maximally helicity violating (MHV) and next-to-MHV (NMHV) amplitudes at six points through six loops [60,61,[68][69][70][71][72][73][74], and at seven points through four loops [63,75]. While the fact that only multiple polylogarithms show up in these amplitudes remains conjectural, there exists evidence that it holds to all loop orders [5]. Also, the specific arguments of the polylogarithms in the six-point case are consistent with a recent all-orders analysis of the Landau equations [76].
In this article, we will focus on a particular class of DCI integrals inside the space of Steinmann hexagon functions. These integrals have a 'double pentaladder' (hereafter just 'pentaladder') topology, meaning they take the form of a ladder integral capped on each end by a pentagon with three external massless legs, for a total of six massless legs. Starting at two loops, there are two integrals with this topology, denoted by Ω (L) andΩ (L) , corresponding to two inequivalent numerator factors that render the pentagon integration infrared finite. These integrals constitute the most nontrivial part of the amplitude at two loops, and contribute to the amplitude at all loop orders [3]. Moreover, members of these classes of integrals are known to be related to each other at adjacent loop orders by a pair of second-order differential equations [70,77]. Armed with these differential equations, we consider finite-coupling versions of Ω (L) and Ω (L) by summing over the loop order weighted by (−g 2 ) L , as was done previously for a related box ladder integral [78]. While these quantities are not the full finite-coupling six-point amplitude, they do constitute well-defined contributions to it that sum up an infinite class of Feynman integrals. By exploiting the symmetries that preserve the dual coordinates on each side of their ladders, variables can be found that simplify the differential equations these integrals obey. Remarkably, after performing a separation of variables, we obtain compact representations of the finite-coupling versions of Ω (L) andΩ (L) in terms of a single Mellin integral over products of hypergeometric functions. These representations are valid for any value of the coupling. Factoring the second-order differential operators into first-order operators, we are led to consider two additional classes of integrals, O (L) and W (L) , that inherit this finite-coupling description. In order to generate more Steinmann hexagon functions, we go on to consider the enveloping space of polylogarithmic functions that is generated by taking all possible derivatives of these integrals at arbitrarily high loop order. We refer to this space of functions as the Ω space. It is graded by an integer weight, where for example Ω (L) has weight 2L. Surprisingly, the nontrivial part of the Ω space is entirely encoded in the discontinuity of these integrals with respect to the channel carrying momentum along the ladder. After taking this discontinuity, the finite-coupling representation of each integral can be rewritten as a contour integral over a branch cut that collapses to a pole in the weak coupling expansion. Perturbatively, the dependence on the kinematic variables reduces to powers of logarithms in one variable and single-valued harmonic polylogarithms (SVHPLs) [79] in the other two variables. This simplicity allows us to recursively construct the function space corresponding to this discontinuity to arbitrary weight. Promoting this space to the full Ω space also turns out to be incredibly simple, since the kernel of the discontinuity operation within that subspace contains only two functions at each weight.
Using similar methods, we also resum the pentabox ladder integrals, which are capped by a pentagon on one end of the ladder and an off-shell box on the other end. These integrals contribute to seven-and higher-particle amplitudes in planar N = 4 SYM. There is an analogous enveloping space of polylogarithmic functions associated with these integrals, which can be easily constructed by taking a kinematic limit of the Ω space.
These new finite-coupling representations give us formidable control over the original Lloop integrals. In various kinematical limits, they lead to explicit formulae for the integrals to high loop orders. They also give us a handle on the structure of the Ω space, which (as a space of multiple polylogarithms) is endowed with a Hopf algebra and an associated coaction [80][81][82][83][84][85][86]. In particular, the relevant discontinuities of the Ω (L) ,Ω (L) , O (L) , and W (L) integrals are related to each other by first-order differential operators, and this system of differential equations is encoded in the coaction. The coaction can thus be realized as a 4 × 4 matrix that acts on the vector of the discontinuities of these integrals. (The coaction on the integrals themselves maps to a slightly larger space of functions, and must be described by a larger matrix.) The coaction on the discontinuity of these integrals can also be defined at finite coupling (that is, nonperturbatively) in the form of a matrix product of path-ordered exponentials. By construction, this nonperturbative coaction satisfies a coaction principle [87][88][89], meaning that the first entry of the coaction always maps to the original space of discontinuity functions, while the second entry can map to a larger space, in general. We expect that this structure can be lifted to the full Ω space. A similar coaction principle also seems to be at work in perturbative string theory [90,91], φ 4 theory [89], QED [92], and the full space of Steinmann hexagon functions (where data currently exist through six loops) [69,74]. The finite-coupling structure of the Ω space lends weight to the conjecture that H is endowed with a similar structure to all orders. In many ways, the Ω space thus serves as an instructive toy model for the full space of hexagon functions, as well as for quantum field theory more generally.
In studying these integrals, we hope to inaugurate a new approach to Feynman integrals that goes beyond order-by-order progress in perturbation theory. The Ω (L) andΩ (L) integrals can now be described analytically to any order, as well as at finite coupling -the already substantial all-orders understanding of this class of integrands is now complemented by a thorough understanding of the type of functions to which they integrate. Similar types of functions can be expected to appear in planar N = 4 more generally, at least in the MHV and NMHV sectors. We hope that other infinite families of integrals can be identified and characterized in a similar manner, eventually extending such an all-orders description to scattering amplitudes themselves.
The remainder of this paper is organized as follows. In section 2 we define the Ω (L) and Ω (L) integrals. We also define the related pentabox and box ladder integrals, for which the pentagon at one or both ends of the ladder is replaced by an off-shell box. We then introduce the differential equations these integrals satisfy. In section 3 we leverage the symmetries of the Ω (L) integrals, as well as their single-valuedness in the region where all cross ratios are close to 1, to resum them into a one-fold Mellin integral over hypergeometric functions. We do the same forΩ (L) , and introduce a family of related functions. In section 4 we show that the finite-coupling representations of these integrals may be equivalently recast as an infinite series, corresponding to the Taylor expansion around a particular kinematic limit. By further expanding the coefficients of this series at weak coupling, we may resum it away from the limit for certain two-dimensional slices of the space of kinematics, in terms of multiple polylogarithms. Then, in section 4.6 we analyze our integrals at strong coupling, finding evidence that they become exponentially suppressed for a large chunk of the Euclidean region.
Section 5 describes the Ω space of functions appearing in the coaction of the basic integrals. It showcases a space of functions relevant to the six-point scattering amplitude that can be constructed explicitly to all loop orders. First we consider the discontinuity of the functions with respect to the channel carrying momentum along the ladder. This discontinuity is simpler to analyze, yet it contains nearly all the information about the full space. In particular, the discontinuity space can be efficiently reconstructed from its coaction, which we formulate nonperturbatively. We conclude in section 6 with a discussion of these results and possible directions for future work.
This paper includes three appendices: Appendix A collects relations between different sets of kinematic variables; appendix B describes some "extended Steinmann relations" that have been found in the full space of hexagon functions H; and appendix C describes analogous relations for the spaces Ω and Ω c , as well as coproduct relations between the various integrals, and how a curious "double coproduct" operator acts on the Ω space. We also provide three ancillary files. Two of them, omega1vwL0-8.m and omegauv0L0-8.m, give the integral Ω (L) on the surfaces u = 1 and w = 0, respectively, through eight loops in terms of multiple polylogarithms. The third, omegacdiscwt0-12.m, gives the c-discontinuity of all the functions in the Ω space through weight 12.

Dual conformal symmetry
In addition to the superconformal symmetry that follows from its Lagrangian formulation, N = 4 SYM develops a dual conformal symmetry in the planar limit [51][52][53][54][55]. This new symmetry is associated with conformal transformations acting on the dual (or region) coordinates where p αα i is the momentum of the i th scattering particle, and x αα n+1 ≡ x αα 1 . Planarity implies that these coordinates can only appear in integrals via the squared differences where x 2 i,i+1 = 0 when leg i is massless. The planar loop integrand also depends on dual coordinates x r , x s , etc., associated with the interior region of each loop. After dividing out by the tree-level MHV superamplitude, the loop integrand multiplied by the integration measure becomes dual conformal invariant [55,93]. In particular, such an object must be invariant under the dual conformal inversion operator I, As a consequence, external dual coordinates should appear the same number of times in the numerator and denominator of the integrand, while the dual loop coordinates should appear four more times in the denominator. For example, the two-loop pentaladder integrals can be written in terms of the integral where, in addition to dual coordinates associated with each loop, we have introduced a pair of points x αα a and x αα b that solve the null-separation conditions [77]. This choice suppresses the integrand in each of the limits where the denominator vanishes, rendering the integral infrared (IR) finite.
These conditions each admit two parity-conjugate solutions, namely , (2.5) where ij ≡ αβ λ α i λ β j and [ij] ≡ αβλα iλβ j . These solutions seem to give us four possible choices of numerator in the pentaladder integral (2.4). However, only two of these choices give rise to different integrals, because replacing both x αα a and x αα b with their parity conjugates gives rise to integrals that differ only by terms that vanish after integration [77].
Choosing pairs (x a 1 , x b 1 ) or (x a 2 , x b 2 ) that are related by the cyclic shift i → i + 3 gives rise to the integral Ω (2) (up to a kinematic prefactor required to make it DCI, given below). On the other hand, choosing the pair (x a 1 , x b 2 ) gives rise to the integralΩ (2) (again, up to a kinematic prefactor), which is parity conjugate to the integral (x a 2 , x b 1 ).
These integrals can also be expressed in terms of momentum twistors [94,95] where R = (α,α) is a combined SU (2, 2) index. Momentum twistors live in the projective space CP 3 , as they are invariant under the independent little group rescalings, Z i → t i Z i for each i. They are related to the squared differences defined in eq. (2.2) by , but is not projectively invariant. The spinor products ij are not DCI, but cancel out in projectively-invariant, DCI ratios. Using eq. (2.8), the expression for Ω (2) can be written in terms of momentum twistors as which has been made projectively invariant and DCI by the inclusion of the kinematic factor 1256 2345 6134 . The modified index structure of the integration variables, from single dual indices to pairs of momentum twistor labels, encodes the fact that points in dual space map to lines in momentum twistor space. We have additionally used the replacements x a → Z 1 Z 3 , x b → Z 4 Z 6 , which selects out x αα a 1 and x αα b 1 from eqs. (2.5) and (2.6). Notice that this integral is invariant under the dihedral transformation that exchanges momentum twistors (legs) 1 ↔ 3 and 4 ↔ 6 while leaving 2 and 5 invariant.
To writeΩ (2) in the language of momentum twistors, we use the fact that parity maps Z i to the ray orthogonal to Z i−1 , Z i , and Z i+1 . In particular, mapping Z i and Z j to their parity conjugates sends the four-bracket where (i − 1ii + 1) ∩ (j − 1jj + 1) denotes the intersection of the hyperplanes spanned by {Z i−1 , Z i , Z i+1 }, and {Z j−1 , Z j , Z j+1 }, respectively. Using this map to send x αα b 1 to x αα b 2 in eq. (2.9), we arrive at the expressioñ where we have made use of the identity 11) and have replaced the previous kinematic factor by 1246 2346 . (We have additionally multiplied by an overall minus sign to stay consistent with the definition in the literature [70].) Unlike Ω (2) , which is parity even,Ω (2) has both a parity even and parity odd part. Like Ω (2) , it is symmetric under the dihedral transformation 1 ↔ 3, 4 ↔ 6.
x 1 x 3 x 5 x 7 Figure 1. The eight-or higher-point L-loop ladder integral, labelled by dual coordinates.

The Box Ladder, Pentabox Ladder, and (Double) Pentaladder Integrals
Before introducing the L-loop generalizations of Ω (2) andΩ (2) , let us first consider the simpler 'box ladder' integrals shown in fig. 1. The box ladder integrals involve only four dual coordinates, but none are null separated, so the first all-massless scattering amplitude to which they could contribute would be an eight-point amplitude.
The representative of this class at one loop is just the DCI four-mass box integral, . (2.12) The L-loop integral can be defined iteratively by integrating over the appropriate dual coordinate in the (L − 1)-loop integral, weighted by the appropriate propagator factors. The box ladder integrals depend on only two cross ratios, conventionally expressed in terms of the variables z andz defined by wherez = z * on the Euclidean sheet where the cross ratios are real and positive. In general, for a sequence of L-loop ladder integrals I (L) , we define the finite-coupling, or resummed, version by i.e. we just drop the (L) superscript. Typically the integral can be normalized so that I (L) is a pure function, that is, an iterated integral with no rational prefactor, for L ≥ 1, while the tree quantity I (0) is rational.
x 3 x 4 x 6 x 1 x 2 Figure 2. The seven-or higher-point pentabox ladder integral, labelled by dual coordinates, where the ladder is formed out of L − 1 loops. The dashed line represents a numerator factor that renders the integral DCI.
The box ladder integrals have long been known to all loop orders [96,97], and can be written as [96,98,99] This class of integrals has been evaluated at finite coupling, i.e. resummed to all orders [78]: (2.18) Here we have changed normalization by multiplying the result given in eq. (21) of [78] by tµ/i, and changed integration variables from z to ζ to avoid confusion with our variables z andz (in our variables, φ = arg z). Finally, we have taken κ 2 → 4g 2 in order to match our coupling and normalization conventions. This result will provide a cross-check of the method used to obtain our main finite-coupling results in section 3.
We can promote the ladder integrals to the pentabox integrals shown in fig. 2 by attaching a pentagon to the end of one of the ladder integrals. This is done by carrying out a single integration on the ladder integral of the form where the point x a should be chosen to be null-separated from the dual variables x 1 , x 2 , x 3 , and x 4 . Choosing x a 1 or x a 2 in eq. (2.5) gives the same result, i.e. this integral is parity-even.
The pentabox ladder integrals involve five dual coordinates. Since two pairs of coordinates are not null separated, the first all-massless scattering amplitude they can appear in is a seven-point amplitude.
The pentabox ladder integrals can be alternatively defined by attaching L − 1 boxes to the one-loop pentagon integral, where the boxes are added iteratively by the integration The pentabox ladder integrals depend on the cross-ratios . (2.22) Our main interest in this paper is in the (double) pentaladder integrals, which involve six dual coordinates, all null separated from their neighbors, so that these integrals will appear in all-massless six-point amplitudes.
There are two classes of pentaladder integrals that can be defined, corresponding to the two inequivalent numerator choices highlighted in the last section. The diagram for the first class of integrals, Ω (L) , is shown in fig. 3. The dashed lines in this diagram indicate the numerator factors x αα a 1 and x αα b 1 , although we could have equivalently chosen x αα a 2 and x αα b 2 (which, in our convention, would have swapped these dashed lines for wavy lines). The diagram forΩ (L) differs by the exchange of just one of these numerator factors for its parity conjugate-or, graphically, by the exchange of one of the dashed lines for a wavy line.
These pentaladder integrals can be most easily defined in momentum twistor space, as repeated insertions of a box into Ω (2) andΩ (2) . For example, the three loop integrals may be obtained from eqs. (2.9) and (2.10) by the replacement with an obvious generalization to higher loops. Six-point DCI integrals can in general depend on the three cross ratios (2.24) As at two loops, Ω (L) andΩ (L) are both symmetric under the simultaneous exchange of legs 1 ↔ 3 and 4 ↔ 6. This transformation exchanges u and v, but we will have to be careful about signs when transforming the parity-odd part ofΩ (L) . The distinction between these integrals in terms of their numerator factors follows the same rule as at two loops-namely, Ω (L) picks out pairs of points x a and x b that are related by the cyclic shift i → i + 3 in their dual indices, while forΩ (L) these points are related by i → i + 3 plus parity conjugation.
x 3 x 4 x 5 x 6 x 1 x 2 Figure 3. The six-point integral Ω (L) , labelled by dual coordinates, where the ladder is formed out of L − 2 loops. The dashed lines represent numerator factors that render the integral DCI.

Climbing the ladders with differential equations
These ladder integrals all share one crucial attribute: they satisfy a set of differential equations that relate adjacent loop orders [77]. For the box ladders, this differential equation is quite simple to write down, and has been known for some time: We have rearranged the traditional presentation of this relation, which relates f (L) to f (L−1) , to the all-loop expression f (z,z, g 2 ) in order to emphasize that it really is valid for finite coupling. A similar, if slightly more complicated, differential equation applies to the pentabox ladders. We define the quantity and its finite-coupling version Ψ(u, v, g 2 ) via eq. (2.15). Then Ψ(u, v, g 2 ) obeys the differential equation Each of the (double) pentaladders also satisfies a differential equation, which we first give at fixed loop order L. The differential equation relating Ω (L) and Ω (L−1) was derived in momentum-twistor space, and reads [77] 1234 2345 6134 The corresponding relation betweenΩ (L) andΩ (L−1) is This equation was given in ref. [70] for L = 2, but the same derivation holds for any L > 2, except that the sign of the right-hand side needs to be flipped; we also redefineΩ (1) → −Ω (1) with respect to ref. [70]. Note that Ω (1) andΩ (1) are one-loop hexagon integrals with double numerator insertions [70,93]. We define the finite-coupling versions of Ω (L) andΩ (L) by, The finite-coupling analogs of the above differential equations will be discussed in section 3, after we introduce some new kinematic variables which dramatically simplify them.

Separated form of the differential equations
The ladder in the Ω (L) andΩ (L) integrals is framed by the dual coordinates x 1 and x 4 , as can be seen in fig. 3. We will exploit the symmetries that preserve these two points in order to write a finite-coupling expression for these integrals. The same technique will also be applied to other systems containing the same ladder. Thanks to dual conformal symmetry we can put x 1 and x 4 at zero and infinity, respectively. It is then easy to see that the symmetry preserving their location consists of SO(4) rotations and scale transformations. Writing the SO(4) algebra as a product of two SU(2)'s, this indicates that the ladders are controlled by a SU(2) L ×SU(2) R ×GL(1) symmetry. The idea will be to find variables which transform as simply as possible under each factor of this group.
Let us first parametrize the hexagon kinematics explicitly in this frame using the embedding formalism. Each dual coordinate x µ i ≡ σ µ αα x αα i (where σ are the usual Pauli matrices) is encoded as a null six-vector where i labels the columns of the matrix. Here we have put the points X 1 and X 4 at 0 and ∞, respectively. Because the six external momenta in the ladder integral are massless, in addition to X 2 i = 0 for each i, we also have that X i · X i+1 = 0. This forces many components to vanish, and implies in addition that p 2 i = 0. Note that the p i are related to, but not equal to, the momenta in the original frame.
The ladder integrals depend only on the cross ratios (2.24), which now evaluate to Notice that these variables are invariant under separate rescalings of p 4 and p 5 , and are also invariant under a common rescaling of p 1 and p 2 . Let us now look at the action of the SU(2) L ×SU(2) R ×GL(1) symmetry on one endpoint of the ladder, that is, on X 2 and X 3 , which according to eq. (3.1) depend on p 2 and p 4 . The GL(1) scale transformations act simply by rescaling p 2 . Using the spinor helicity factorization for null vectors, p i = λ iαλiα , the first SU(2) factor acts on holomorphic spinors λ 2α and λ 4α , and the second on anti-holomorphic spinorsλ 2α andλ 4α . This motivates introducing the following variables: To find the change of variables between (u, v, w) and (x, y, z), first note from eq. (3.2) that which readily yields Using the Schouten identity in eq. (3.3) one similarly finds Solving eqs. (3.5) and (3.6) for x and y in terms of u, v, w, we get The choice of sign in front of √ ∆ in eq. (3.7) is somewhat arbitrary; parity flips this sign and exchanges x ↔ y.
The differential equations (2.28) and (2.29) are expressed in terms of momentum twistors, so it is useful to write x, y, z in terms of ratios of momentum-twistor four-brackets. In appendix A we recall the momentum-twistor representations of the cross ratios u, v, w, and also of the variables y u , y v , y w . Both u, v, w and x, y, z are rational functions of y u , y v , y w ; see eqs.   In this representation, it is easy to show that the dihedral flip Z i ↔ Z 4−i that leaves Ω (L) andΩ (L) invariant transforms the above variables as x ↔ y and z → 1/z. Also notice that under the cyclic transformation Z i → Z i+3 , x is exchanged with y, while z is left invariant, allowing us to identify this transformation with parity. This transformation also sends the dual coordinates x i → x i+3 . Inspecting fig. 3, we see that Ω (L) is invariant under parity because the left and right numerator factors transform into each other under i → i + 3. In contrast,Ω (L) has both parity-even and parity-odd parts. The x, y, z variables simplify the momentum-twistor differential operators appearing in eqs. (2.28) and (2.29). Using the Schouten identity for four-brackets and the chain rule, we have Using these relations, the differential equation (2.28) for Ω becomes − 2345 1345 which can be expressed in terms of y and z only as At finite coupling, using the definition (2.30), the Ω ladders thus satisfy the equations: where we have also used the fact that Ω(x, y, z, g 2 ) is even under parity, x ↔ y, to add the second equation. This form of the equations will be very convenient because they now take on a separated form, thanks to switching to the x, y, z variables. The corresponding differential equations forΩ(x, y, z, g 2 ), derived in a similar way from eq. (2.29), are In particular, expressing (2.29) in terms of the x, y, z variables leads to (3.15), whereas (3.16) follows from the latter by a flip transformation, as discussed under (3.8). Notice from the momentum-twistor forms (2.28) and (2.29) that the second-order part of the differential operator is exactly the same for Ω andΩ, because the 2345 and 2346 factors cancel in the second-order terms. The extra linear term forΩ arises when the operator Z 1 · ∂/∂Z 2 acts on the factor 2345 / 2346 .

Pentaladders
We now turn to the solution of the differential equations (3.13) and (3.14) for Ω, and (3.15) and (3.16) forΩ. We begin by diagonalizing z∂ z using a Mellin representation. (A related double Mellin representation of the box ladder integral has been obtained using integrability [98].) We seek separated solutions for Ω(x, y, z, g 2 ) of the form Equation (3.14) then gives while eq. (3.13) gives the identical equation for F in y. The four independent solutions to the pair of differential equations can be labeled by the signs of ν: where F j ν are hypergeometric functions, normalized to F j ν (1) = 1: Below when discussing the box ladders we will find that j(ν) − 1 is the SO(4) spin, which suggests viewing the differential equations (3.13) and (3.14) intuitively as two relations among the three Casimir invariants of SU(2) L ×SU(2) R ×GL(1).
To find the physically relevant combination of the solutions (3.19), we impose the fact that Ω must be smooth in the entire positive octant u, v, w > 0. In particular, consider the neighborhood of the point (u, v, w) = (1, 1, 1), where x and y are both small. The function should admit a regular Taylor series expansion. However only the combination xy is regular: x/y depends in a complicated way on the angle of approach. From the behavior of eq. (3.20) near the origin, we see that requiring the leading term in this limit to be a power of xy leaves only two acceptable solutions: F To get a further constraint, we note from eqs. (3.5) and (3.6) that: This singles out a unique linear combination of the above two functions. We conclude that an integral representation of the following form must hold: where we have integrated over the dilatation eigenvalue ν with a yet undetermined coefficient c(ν, g 2 ). The insertion of the explicit factor of 1/ sinh(πν) is motivated by the following consideration: Regularity of the Taylor series at (u, v, w) = (1, 1, 1) implies that the singularities of c(ν,g 2 ) sinh(πν) can be at most single poles at imaginary integers, since in this limit x, y → 0 and the integral can be done by residues (closing the contour below the real axis on the first term, and above in the second term). Thus c(ν, g 2 ) is an entire function.
To fix this function, we find a boundary condition at large ν. At large ν, the integral simplifies dramatically, because one of the indices on the hypergeometric functions, iν − j, goes to zero, so using eq. (3.5) we get simply We see that for the above term, the large ν region dominates 2 the limit v → +∞. Choosing the phase, say, the integrand numerator acquires a factor e πν , canceling the same factor from the sinh(πν) in the denominator and making it marginally convergent. Thus If c approaches a constant, we get linear growth in v as v → ∞. Now, at tree level (g 2 = 0), we have that Ω(u, v, w, 0) = 1 − u − v. (We didn't define Ω (0) as an integral but it can be found by acting on the known one-loop integral Ω (1) with the differential operator (3.14).) Hence the tree-level quantity does grow linearly in the limit. On the other hand, the loop corrections Ω (L) with L ≥ 1 grow at most logarithmically in this limit, since they are uniform transcendental functions. We conclude that the missing coefficient must be independent of the coupling, and equal to c(ν, g 2 ) = 1/(2i). Our final result is therefore: where x, y, z are defined in eqs. (3.5) and (3.7), j(ν) = i ν 2 + 4g 2 , and the hypergeometric functions F are defined in eq. (3.20). This formula is the main result of this section.
As a simple check, at tree level, g 2 = 0, the hypergeometric functions become trivial and F Performing the integral by closing the contour in the lower half-plane and summing over residues at ν = −ik, we reproduce the result Ω(u, v, w, 0) = 1 − u − v. In general, it is straightforward to expand the integral (3.26) around (u, v, w) = (1, 1, 1), since x and y are both small and so only the residues at a finite number of poles contribute. The residues give hypergeometric functions and only a finite number of terms in their Taylor expansions are needed. The expansion coefficients can therefore be obtained exactly in g 2 . At one loop, we have resummed the series around (1, 1, 1), finding as expected: (3.28) Section 4 discusses how to perform such expansions at higher loop orders.
The pentaladder integralΩ with mixed numerators can be analyzed in an identical fashion. The only change is that the solutions to the differential equation (3.16) with the separated form (3.17) instead of eq. (3.20). These functions are to be multiplied by the solutions to (3.15), which are found (by letting x → y and ν → −ν) to be F j(ν) −ν (y) and F j(ν) ν−2i (y). As in the case of Ω, imposing regularity in the positive octant fixes a unique combination of these solutions. It first requires the pairings F j(ν) −ν (y), and then it imposes a relative minus sign between the two.
We also require that the large-v limit is saturated by its tree-level expression. The oneloop result [93] isΩ (1) normalized with the opposite sign as eq. (E.4) of ref. [70]. We note that this is not a pure transcendental function, althoughΩ (L) is pure for L ≥ 2. The integralΩ (L) has only been defined so far for L ≥ 1 but we can define it at L = 0 by applying the differential equation (3.15), which gives us, after simplification using formulas in appendix A: This expression has a strange form for a "tree" object, since it contains a logarithm. However, hittingΩ (0) once more with the differential operator gives zero, confirming that it is indeed the leading term of theΩ family.
In the limit of large v (choosing the branch of the square root where x → 1, corresponding to y v → 0, y u → ∞ with y u y v → y), we have 1−yv 1−yu → 0. HenceΩ (0) is dominated by the first term, which grows linearly like v(1 − u). The one-loop result (3.30) and the higher loop orders grow at most logarithmically, so the large-v limit is again tree-level exact. Matching the integral representation with this limit, our final result forΩ is .
We note the asymmetry between x and y, which originates from the differential equations (3.15) and (3.16):Ω is not invariant under parity.

Differential relations among pentaladder integrals
It turns out that the functions Ω andΩ are not independent, but rather they appear as derivatives of one another. These relations can be understood as properties of the hypergeometric functions entering eqs. (3.26) and (3.32). It will prove useful to first introduce other auxiliary integrals, O(u, v, w, g 2 ), which we call the odd ladder because it has odd parity and its perturbative coefficients O (L) all have odd weight, and an even companion W(u, v, w, g 2 ). The odd ladder will be a generalization of the one-loop six-dimensional integralΦ 6 studied in ref. [100]. (See also ref. [101].) The latter integral satisfies and it is the first parity odd function in the space of hexagon functions. By analogy with eq. (3.33), we now define using eq. (A.13) and identifying O (1) = −Φ 6 . The corresponding finite-coupling definitions of O and its even companion W are This normalization of W will prove convenient in section 5. Given the finite-coupling solution for Ω, eq. (3.26), we can obtain one for the odd ladder integral simply by acting with the differential operator on the products F j ±ν (x)F j ±ν (y) in the integrand. The hypergeometric function satisfies which can be verified, for example, using the hypergeometric function series representation (4.2). Note that here we used that −ν 2 − [j(ν)] 2 = 4g 2 .
When we apply eq. (3.36) to eq. (3.35), using eq. (3.26), the iν/2 terms cancel, and we are left with: Similarly, We are now in a position to discuss several first-order differential relations among the integrals. If we take the difference between eq. (3.13) and eq. (3.14) so as to cancel the z derivative, then the x and y derivatives factorize as (x∂ x ) 2 − (y∂ y ) 2 = (x∂ x + y∂ y )(x∂ x − y∂ y ), which shows that Ω can also be written as a first derivative of the odd ladder integral, This equation generalizes a relation found between Ω (1) and a derivative ofΦ 6 in ref. [100]. However, we also find empirically, to high orders in perturbation theory, that the x and y derivatives of O also contain the parity-even part ofΩ, which we callΩ e , and the z derivative of O generates the parity-odd part,Ω o , wherẽ (3.40) In particular, we find that which gives back eq. (3.39) and alsõ The parity-odd relation is justΩ These two empirical relations combine tõ In appendix C.3, we provide additional relations of a similar nature. We would like to derive these differential relations from the finite-coupling integral representations. In order to do so, it is useful to consider the derivatives of F If we apply the differential operator (1−x)x d/dx to the left-and right-hand sides of eq. (3.36), we can use the second-order differential equation (3.18) satisfied by F j ν to simplify the lefthand side, obtaining a formula for the derivative of F Furthermore, only two out of the three functions F +ν (x) can be linearly independent; indeed, hypergeometric identities can be used to show that The physical significance of the four functions we have introduced is now clear: they form a complete basis for products of F and F with arguments x and y that are smooth in the Euclidean region. More precisely, we can form the four-vector where the subscript ν means to focus on the ν-integrand (i.e. to drop dν/(2i) in each of the integrals in eq. (3.38), eq. (3.26), eq. (3.32) and eq. (3.37)). Because this basis is complete, all the x, y and z derivatives of the V i can be expressed as linear combinations of the V i , allowing the total differential to be written in matrix form: Computing the derivatives, we find the following explicit form for the matrix M : The matrix M provides us with first-order differential relations between integrals. Since the variable ν is to be integrated over, to find these relations we should take combinations of x and y derivatives that are independent of ν. For example, the x and y derivatives of the second row of M give us back the definitions eq. (3.35), whereas the x and y derivatives of the fourth row give the two nontrivial relations eq. (3.39) and eq. (3.43). Finally, from the first and third row we find two additional first-order relations: The first follows readily from the factorized form of the Ω differential equation. Using eq. (3.44), the second-order eq. (3.15) forΩ could also be readily rederived from the matrix M . This discussion shows that Ω andΩ naturally fit inside a common system. In section 5 we will use the matrix M to define an enlarged set of transcendental functions, which will be closed under the action of taking any derivative.

Pentabox ladders
Consider now the similar integral in fig. 2 where we chop off the pentagon on the right and replace it by a box. The two light-like-separated dual coordinates of the right pentagon are replaced by a single point q = x 6 which is not light-like separated from x 1 or x 4 .
At one loop, for example, the integral we consider is given in eq. (2.20), or in terms of seven momentum twistors, Using the embedding formalism as in eq. (3.1), and again putting the sides of the ladder at 0 and ∞, we parametrize the external kinematics as where q 2 = 0 reflects the external masses on one side of the ladder. The two cross-ratios are now: Scale transformations act by rescaling q, which leaves invariant the variable Therefore we make an ansatz for Ψ as a sum of terms Using the chain rule, we can rewrite u and v partial derivatives in terms of x and z derivatives, Applying also the identity ( which has exactly the same form as eq. (3.14) for Ω(x, y, z, g 2 )! Thus for the ansatz (3.58) we get exactly the equation for F (x) given in eq. (3.18); that is, the partial wave decomposition of Ψ gives a sum of F j(ν) ±ν terms. As in the double pentagon case, here we could again argue that the relative coefficient is fixed by analyticity around u, v = 1 and u + v = 1, up to an entire function, itself fixed from the v → ∞ limit.
In fact the equivalence of the two problems was already understood from the differential equation in ref. [77]: we can get to the pentabox ladders by taking the w → 0 limit of the result (3.26), where y → 1 and thus F j ±ν (y) → 1. This gives immediately: . (3.61)

Box ladders
We can conduct a similar analysis for the box ladders depicted in fig. 1. In this case a finite-coupling expression was already derived in ref. [78]. We will show that we reproduce this expression, which we presented in our conventions in eq. (2.18). We consider the ladder whose long sides are labelled by x 1 and x 5 , as in fig. 1, so again we set these points to zero and infinity. The data then are the ratio x 2 3 /x 2 7 and the angle between x 3 and x 7 , which are the norm and phase of the complex variable z defined in eq. (2.14). The conventional cross ratios u and v are defined by ; (3.62) note that u/v = zz. The sequence of ladders obeys the differential equation: as previously presented in eq. (2.25). The one-loop case is The differential equation then requires Again we make an ansatz for f as a sum of terms of the form: that is, φ = arg z when z is complex, and the differential equation becomes The general solution is a combination of F ±j (φ) = e ±ijφ , with j(ν) = i ν 2 + 4g 2 as before.
Because j(ν) is conjugate to φ, the angle between x 3 and x 7 , it should be interpreted as SO (4) spin. 3 To find the correct combination of solutions F ±j (φ), we wish to impose that the loop corrections to f vanish when z =z, as a consequence of the factor (z −z) removed in eq. (2.16). Working in the Euclidean regionz = z * , this occurs for two different values of the phase: φ = 0, π. As we will see below, the proper interpretation of the vanishing at φ = 0 turns out to be subtle, because there is a singularity at z =z = 1. In fact the restriction to φ = 0 at tree level results in a nonvanishing distribution supported at that point. However, f vanishes identically at φ = π. This means that for each value of ν only the combination of solutions F ±j (φ) that vanishes at φ = π contributes, so that we can write: The symmetries of the integral imply that c(ν, g 2 ) is an even function of ν. Now consider single-valuedness as (zz) → 0. There the ν-integral will be done by residues in the lower-half plane and each residue should correspond to an integer spin j in order to be single-valued. This implies that the only singularities of c(ν,g 2 ) sin(πj) are single poles at integer spin, and therefore c(ν, g 2 ) is an entire function.
To fix the asymptotics of c(ν, g 2 ) we consider the limitz → 1 (with z otherwise fixed). We get a power divergence at tree level, but only logs at loop level; in the above expression this divergence will come from large positive ν, where j ≈ iν: Because c(ν, g 2 ) is entire, this behavior determines it uniquely: We can now understand the vanishing at φ = 0 more precisely. The sine factors cancel in this limit and the integral produces a delta-function δ(|z| − 1) that is independent of g 2 . This reproduces precisely the singular behavior of the tree-level function, and otherwise it vanishes for generic |z|.
We remark that eq. (3.69) resembles an integrability-based representation of the box ladder integral [98], in which the variables analogous to ν are spectral parameters.
Let us compare with eq. (2.18). Setting ν = ζ 2 − 4g 2 , ζ = −ij(ν) in this integral, we get which is precisely the same as eq. (3.69). We conclude that the method works for the box ladders as well, although it involves an additional subtlety because of the tree level singularity at z,z → 1.

Sum Representation and Perturbative Evaluation as Polylogarithms
The integral representations presented in the previous section capture the Ω integrals fully at finite coupling. If one is interested in extracting numerical values, or in finding Ω (L) at a particular loop order, it is useful to derive alternate representations in terms of infinite sums. In this section we will derive a representation of this sort, and use it to efficiently find polylogarithmic expressions for Ω (L) in specific limits.

Sum representation
In eq. (3.26), we may change the sign of the integration variable ν in the term containing F j −ν , and rewrite the integral as where P denotes the Cauchy principal value, which is necessary because now the integrand has a pole on the integration contour, at ν = 0. This simply amounts to the prescription of including half the contribution of this pole to the integral. Using the power series definition of the 2 F 1 Gauss hypergeometric function 4 , we can deduce that F j ν (x) will produce poles when (iν ± j)/2 + n = −k, or equivalently when namely only in the upper-half ν-plane. We may thus choose to evaluate eq. (4.1) by closing the contour in the lower-half plane, picking up poles at ν = −ik from the sinh(πν) factor in the denominator. Redefining we thus arrive at the following series representation of the all-loop Ω, By virtue of eq. (4.2), as well as the following argument transformation formula, we may express the functionsF aŝ With the help of these formulas, and the sum representation (4.5), we may easily obtain kinematic expansions of Ω around x = y = 0 to the desired order. Through eq. (A.10), these expansions are equivalent to expansions in (u, v, w) around (1, 1, 1). In a similar way, we can expand the integrals O,Ω o ,Ω e and W around x = y = 0. For this purpose, we also need the expansion of F j(ν) ν−2i . We define Note thatF j −ik only differs fromF j −ik by a factor of n in the n th term, and an overall factor of 1/g 2 . The two series expansions are related by the latter result following from eq. (4.8).
Then the series expansion of the all-orders odd ladder integral O is: The expansion of the odd part ofΩ can be found by applying −z∂ z to eq. (4.12) : The expansion of the even part ofΩ can be found by expanding the all orders result (3.32) forΩ and then subtracting off the odd part (4.13). The result is Finally, the expansion of the even ladder integral W is given by

Weak coupling expansion
Let us now discuss how to perform the weak coupling expansion of eq. (4.5), as well as its kinematic resummation to multiple polylogarithms, at least in some limits. We begin with the rightmost term in eq. (4.5), coming from the ν = 0 residue where j = 2ig. The F 2ig 0 functions appearing in this term are given by the all-order relation where the factor involving Γ functions in eq. (3.20) was eliminated with the help of the reflection formula, For its weak-coupling expansion, it will be more convenient to use the representation (4.7), together with the identity where are Euler-Zagier sums, or particular values of Z-sums [102]. Inserting eq. (4.18) twice into the series expansion for the 2 F 1 in eq. (4.16), we find After reexpressing the product of Euler-Zagier sums as a linear combination thereof, with the help of the quasi-shuffle (also known as stuffle) algebra relations, for example eq. (4.20) may be immediately evaluated at any loop order in terms of harmonic polylogarithms (HPLs) [103] with argument x, Let us now look at the remaining terms in eq. (4.5). SinceF is symmetric in j ↔ −j, see for example eq. (4.7), the choice of branch when we expand j in the coupling is immaterial, and we may pick Separating the contribution that is small at weak coupling, we may again use the identities (4.17) and (4.18), this time for (k, g 2 ), in order to rewrite eq. (4.7) aŝ This formula allows us to obtain the weak coupling expansion ofF j −ik most efficiently, by first expanding in (k, g 2 ), and then in g with the help of eqs. (4.23)-(4.24). In this manner, it is evident that the most complicated sums in eq. (4.5) will always be of the form To our knowledge, no algorithm currently exists for directly evaluating these kinds of sums in terms of multiple polylogarithms. It would be very interesting to develop one based on our understanding of hexagon functions. However, it turns out that it is indeed possible to resum eq. (4.5) in the limit y → 0 and z → ∞, with x and r = √ xyz held fixed. Inspecting eq. (A.10), we see that this limit corresponds to the following two-dimensional subspace of hexagon kinematics, In this subspace, only the first term in eq. (4.25) survives inF j −ik (y = 0). We let (−r) k = x k × (−r/x) k , replace the summation variable n with n = n + k in the otherF j −ik (x) factor, and exchange the order of summation. Then the most complicated sums take the form ∞ n =1 x n n Z 1,...,1 (n − 1) Crucially, the rightmost sum can be done with the help of algorithm B of ref. [102]. This algorithm has been implemented in the nestedsums library [104] within the GiNaC framework, and by interfacing it with Mathematica we are able to replace all sums in k of the form (4.28) with Z-sums with outer summation index n − 1 (possibly accompanied by rational factors (r/x) n ). With the help of the quasi-shuffle algebra relations, we may rewrite their products with the leftmost Z-sum in eq. (4.28) as linear combinations of Z-sums, similarly to what we did forF 2ig 0 in eq. (4.20). Finally, we evaluate the remaining sum over n in terms of multiple polylogarithms with the help of their sum representation, Very similar techniques have been used to evaluate [105,106] the leading, and part of the subleading, contribution to the hexagon Wilson loop OPE near the collinear limit, as well as to resum [107] all single-particle gluon bound states contributing to the double scaling limit, v → 0 with u, w fixed. The systematic procedure we have described works in principle at any loop order, subject to limitations in computer power. We have used it to obtain explicit expressions for Ω (L) (1, v, w) through L = 8 loops. We quote here the first two loop orders, where Results through eight loops are contained in the ancillary file omega1vwL0-8.m provided with this paper. In precisely the same fashion, we may also resum Ω (L) (u, v, 0), which, as discussed around eq. (3.61), is equivalent to the dual conformal pentabox ladder Ψ (L) (u, v) defined in (2.26) and shown in fig. 2. Starting from the sum representation (4.5), as already mentioned the limit in question amounts to letting y → 1. In this limit, the way F j ν are normalized implies thatF j −ik (y) → 1, also for k = 0. Up to two loops we obtain The polylog arguments are all rational in u, v in this case. Here as well we have carried out the computation up to L = 8; the resulting expressions may be found in the ancillary file omegauv0L0-8.m. The vanishing ofΩ o is also a consequence of its antisymmetry under u ↔ v.
Similarly, the odd ladder integral becomes so it sits in the middle of the differential equation (4.42). On the line (1, 1, w), the even ladder integral is simply related to the odd one at one higher loop: (4.48) As we will see in subsection 4.6, the (1, 1, w) limit offers us insight into the strong-coupling analysis of the integrals. In addition, we can study the radius of convergence in the g plane of the perturbative expansion for Ω(1, 1, w, g 2 ) using eq. (4.38). The same arguments that lead to eq. (4.3) show that the hypergeometric function 2 F 1 (ig, −ig, 1, x) has poles at g = ±i, which are the poles in the g plane closest to the origin. They also match the location of the closest poles of the prefactor πg/ sinh(πg). Therefore the radius of convergence of the perturbative expansion of Ω(1, 1, w, g 2 ) is unity for all w. We can check this result at w = 1 and w = 0 by observing that the ratio of successive loop orders in eqs. (4.39) and (4.40) goes to −1 as L → ∞.
We remark here that the resummed integral Ω(u, v, w, g 2 ) appears correctly weighted in the full BDS-like normalized MHV amplitude E(u, v, w, g 2 ), when g 2 is identified with the standard coupling parameter in planar N = 4 SYM, g 2 = N c g 2 YM /(4π) 2 , where g YM is the Yang-Mills coupling and the gauge group is SU (N c ). In order to establish that it is correctly weighted, one can use the "rung rule" for performing two-particle cuts in planar N = 4 SYM [108,109]. This rules provides the normalization of the Ω (L) terms within the L-loop integrand, relative to the normalization at one lower loop. Therefore it makes sense to compare the unit radius of convergence for Ω(1, 1, w, g 2 ) with the radius of convergence for amplitudes. The latter is not firmly established [68,69,71], but it appears to be closer to 1/16, the value for the cusp anomalous dimension [42], and much smaller than 1. It would be interesting to use our finite-coupling representation (3.26) to investigate the perturbative radius of convergence of Ω(u, v, w, g 2 ) for more general kinematics than just the line (1, 1, w). ) (4.52) HenceΩ =Ω e +Ω o vanishes on the line (1, v, 1), and using eqs. (4.10) and (4.11) we have (4.57) On the line (1, v, 1), the first few orders of explicit results for Ω (L) are:  Although it is not really apparent yet, the coefficients of the ζ 2(k+1) terms for k > 1 are controlled by a 2k-fold convolution of the Catalan numbers. Define which satisfies C n,k = i 1 +i 2 +···+i k =n with C n ≡ 0 for n < 0. It also obeys C n+1,1 = C n+1,2 = C n . Also note from eqs. (4.59)-(4.62) that the Catalan number required is related to the depth, i.e. the length of the (compressed) HPL weight vector. All weight vectors having the same depth appear with the same coefficient, which is nonzero if all entries are ≥ 2. (The only exception is at L = 1, which contains H 1,1 .) We find that, for L > 1, Ω (L) (1, v, 1) is given by where the no-ζ term is and the ζ terms are Here g m,n is the set of weight vectors w = (w 1 , w 2 , . . . , w m ) of depth m and weight m i=1 w i = n, with all w i ≥ 2.
Note that the first term of X (L) 0 also appears in eq. (4.38) for Ω (1, 1, w). Also, using C n+1,2 = C n we see that X , which is exactly the same function describing the no-ζ terms at two lower loops. The ζ 6 terms are the first to require a true convolution of the Catalan numbers, i.e. C n,4 .
Note that the non-Catalan term in Ω (L) (1, v, 1) is equal to the much simpler expression for Ω (L) (1, 1, w), after identifying 1−x = w = 1/v. This is simply the k = 0 term in eq. (4.49), while the terms containing the Catalan numbers come from the k > 0 terms.

Strong-coupling behavior
A remarkable feature of the finite-coupling Ω,Ω integrals (3.26) and (3.32) is that we can evaluate them outside the radius of convergence of the weak-coupling region we used to derive them, all the way to strong coupling. Here we provide evidence that the functions become exponentially suppressed as g → ∞ for a large subspace of the Euclidean domain. This is very similar to the observed strong-coupling behavior of the box ladder integrals for general kinematics [78].
For simplicity, let us begin with the (1, 1, w) line, which we also analyzed in section 4.4 where we focused on weak coupling. At w = 1, the exponential suppression is clear from eq. (4.39): and similarly for w = 0, from eq. (4.40): Ω(1, 1, 0, g 2 ) = − πg sinh πg ∼ −2πg exp(−πg), as g → ∞. (4.72) Generally, going from w = 1 to w = 0 the absolute value of the function increases monotonically between these two limits. To study in detail the behavior between the endpoints, the problem is reduced to the asymptotic analysis of the (normalized) hypergeometric functions F 2ig 0 (x) defined in eq. (3.20), which enters on the first line of (4.38). Fortunately, a very detailed saddle point analysis of this precise class of hypergeometric functions has been carried out in ref. [110], see in particular Theorems 3.1 and 3.2, which are valid in the region x < 0 and 0 < x < 1 respectively. Focusing on the region 0 < x < 1, we can write their result as: which in fact holds for any fixed value of |ν| g, up to relative corrections of order 1/g. The angle φ varies continuously and monotonically from φ(0 + ) = π to φ(1 − ) = 0. Including the second hypergeometric factor in (4.38), which gives F 2ig 0 (0) = πg sinh(πg) , we thus get: The dependence of the exponent on w could also be obtained simply by solving the hypergeometric differential equation at leading order in g. 5 We observe that the exponent smoothly interpolates between eq. (4.71) and eq. (4.72). The prefactors do not quite go smoothly, but this can be understood as a breakdown of the saddle point approximation for extreme values of x very close to the endpoints, see ref. [110] for details. For the other integrals in our basis, it follows from equations (4.47) and (4.48) that O(1, 1, w, g 2 ) and W(1, 1, w, g 2 ) are also exponentially suppressed for all x < 1 (w > 0), whereas we recall thatΩ vanishes identically on this line.
In more general kinematics, we can analyze the integral representation in (3.26), which we reproduce here for convenience: . (4.75) By plotting the integrand for various values of x, y, z we find that it is generally dominated by the region near the origin, where |ν| ∼ 1 g. In this regime we can use the limit in eq. (4.73) to deduce the following exponential suppression of the integrand: where we have focused on the exponent. Thus, assuming that the region |ν| ∼ 1 indeed dominates as suggested by the numerics, Ω is itself suppressed by at least the same factor: Ω(u, v, w, g 2 → ∞) ∼ < e −g(φ(x)+φ(y)) . Remarkably, the true behavior of Ω for generic x, y appears to be even more strongly suppressed. This can be seen from the fact that the asymptotic expansion (4.73) to all orders in 1/g turns out to contain only even powers of ν, so there is a cancellation between the two terms in the numerator of (4.75), causing the dominant behavior to come from subleading exponential corrections to (4.73). While these could in principle be analyzed using formulas in ref. [110], we will simply conclude this subsection with the observation that Ω is exponentially suppressed at strong coupling.

The Ω Space
In this section we analyze the Ω integrals in general kinematics, working perturbatively in the coupling. We define the Ω space of functions to be that containing all iterated derivatives (more precisely, all iterated {n−1, 1} coproducts) of the W, Ω,Ω and O integrals to arbitrary loop orders. We are interested in studying the Ω space primarily as a model for the full space of Steinmann hexagon functions [69,74]. If we can characterize the functions that appear in the Ω integrals to all orders, we will have encompassed a substantial slice of the full space of Steinmann hexagon functions, and this may give hints as to their overall structure. We will achieve this by first showing that a certain discontinuity of the Ω integrals is simple, then using this insight to build the full Ω space.

Coproduct formalism and hexagon function space
Like MHV and NMHV amplitudes in planar N = 4 sYM, the integrals defined in section 2.2 are expected to evaluate to multiple polylogarithms. This implies that they are endowed with a Hopf algebra. In particular, there is a coproduct operation which breaks functions apart into simpler ones. This yields various concrete representations as iterated integrals. For a more complete review of how Hopf algebras and the coproduct show up in amplitudes, see ref. [111]. The key property for us will be that derivatives only act on objects appearing in the second entry of the coproduct ∆: 6 In particular, we can define a set of functions F s by the coproduct action for any multiple polylogarithm F of weight n that involves symbol letters in the set S. The functions F s are then iterated integrals of weight n − 1. The derivative of F with respect to an underlying kinematic variable z (say) is By repeating this operation n times and integrating along various paths with appropriate boundary conditions, one obtains concrete integral representations of F . The coproduct unifies these representations in a canonical way.
In six-particle kinematics, the traditional symbol alphabet is given by [60,83] The list of iterated integrals with this alphabet at any given weight is finite, and is called the hexagon function space. However, constructing this space is nontrivial and currently unsolved beyond weight 12, even after imposing the Steinmann conditions [69] and further restricting to the level of the symbol. For our discussion it will be convenient to parametrize the same space with a new alphabet where a = u vw , m u = 1−u u , and the others are defined by cyclic permutations of u, v, w, as in eq. (A.16). The letters a, b, c are physically significant due to the Steinmann relations, which state that amplitudes (or individual Feynman integrals) can't have simultaneous discontinuities in overlapping channels. Each contains a single three-particle Mandelstam invariant: a ∝ (x 2 25 ) 2 , b ∝ (x 2 36 ) 2 , c ∝ (x 2 14 ) 2 (see the definitions of the cross ratios in eq. (2.24)). In the new alphabet the Steinmann relations state simply that a can never appear next to b in the first two entries of the symbol (or b next to c, or a next to c) [74]. We discuss this condition further in appendix B.
From the kinematic relations (A.10) and (A.16) in appendix A, we see that five of the nine hexagon letters can be taken to be simple combinations of the variables x, y, z which simplify the ladders: These five letters are equivalent under multiplication to which are the only letters appearing in the matrix M introduced in eq. (3.51). As we will now show, after taking a discontinuity in c, the ladder integrals collapse to a space with just these five letters, which will enable their complete description.

The box ladders and their discontinuities
It is helpful to first describe the analogous, simpler, space for the box ladder integrals. The pure functions entering these integrals are given explicitly in terms of classical polylogarithms in eq. (2.17). An alternate representation, which exposes their coproduct structure a bit better, is in terms of single-valued harmonic polylogarithms (SVHPLs) [79].
Recall that the ordinary HPLs [103] with uncompressed arguments w, w i ∈ {0, 1}, obey the differential relations, along with the "initial conditions" The SVHPLs, L w , w i ∈ {0, 1}, are functions of z andz that are linear combinations of products ∼ H w (z)H w (z) and are real analytic in the complex plane minus the punctures at 0, 1, ∞. They satisfy a similar set of differential relations, along with the single-valuedness requirement that the first entry is either zz or (1 − z)(1 −z).
In terms of the L w functions, the box ladders become [99,112], These are depth 1 SVHPLs [113], where the single "1" in the weight vector appears in one of the two middlemost slots. Now we ask, what is the f space of functions that contains all coproducts of the f (L) (z,z) as L → ∞? This is a minimal space in which we can construct all f (L) 's as iterated integrals. Taking the derivative z∂ z simply clips off the first "0". Taking the anti-holomorphic derivativē z∂z can be a bit complicated for a generic SVHPL, but at depth 1 it just clips off the last "0". Iterating this procedure, the "1" can slide forward and backward to any location in the string, until it reaches either end, where it is clipped off by taking a 1 − z or 1 −z coproduct instead of a z orz one. In summary, by taking coproducts using the letters in S f , we generate  table 1 shows what happens to the dimension if we take the discontinuity in 1 − z, holding 1 −z fixed, for all functions in the f space. This discontinuity is associated with the cut in the channel carrying momentum along the ladder [99]. It will be analogous to the c-discontinuity of the pentaladder integrals. The discontinuity of f (L) (defined as the difference of its value for z > 1 taken above or below the branch cut) is given by It is depth 0, since the discontinuity removed the "1". That is, the symbol entries belong to The discontinuity Disc f (L) is itself not a single-valued function. Because of this, when we take derivatives to fill out the full Disc f space, using the fact that discontinuities commute with derivatives, we get all n monomials at weight n − 1: ln k z ln n−1−kz , k = 0, 1, . . . n − 1.
Thus the dimension of the space f is reduced by only one in passing to Disc f , even though the symbol alphabet is halved in size. This means there is very little loss of information in going to Disc f : any function in f can be recovered from its discontinuity at the price of a single boundary condition. Indeed the only combinations of SVHPLs with vanishing z = 1 discontinuity are the simple logarithms ln k (zz).
The ladder integral f (z,z) can be characterized as the unique combination of SVHPLs with the discontinuity (5.14) and which vanishes at z,z → 0.
Before we discuss the pentaladders, it is instructive to understand this simplification from the perspective of the integral representation in eq. (3.69). We fixz to a value between 0 and 1 while we analytically continue to z > 1. The argument of the sine is then complex, π−φ ≡ π− 1 2i ln(z/z), with a real part that approaches ±π depending on whether we approach the cut from above or from below. The discontinuity thus gives a simpler integral with the sin(πj) denominator canceled: The integrand is now an entire function of ν. This is so despite the branch points of j(ν) = i ν 2 + 4g 2 at ν = ±2ig, because it is even in j. It is convenient to use this property to shift the contour slightly below the two branch points. One can then integrate over the two terms separately: closing the contour in the upper-half plane for the first term, where it decays, and below for the second term. The discontinuity thus reduces to a small contour integral over the cut in the first term: letting s = iν/2, It is now easy to see why the discontinuity involves only powers of ln z and lnz in perturbation theory: both exponents are uniformly small over the integration contour, and so can be seriesexpanded. Expanding the integrand at small coupling and taking the coefficient of (−g 2 ) L , we indeed find a pole in 1/s whose residue reproduces eq. (5.14). 7 Notice that an alternative evaluation of eq. (5.16) would have been to move the contour to slightly above the two branch points, closing it in the lower half-plane, and picking up the contribution from the second term in eq. (5.16) instead. In this version, we would have the same formula with j → −j and an opposite overall sign from the reversed contour orientation. Thus eq. (5.17) must be odd under j = 2 s 2 − g 2 → −j. Since this symmetry exchanges z andz, it is consistent with eq. (5.14) being odd under z ↔z.

The pentaladder integrals and their c-discontinuity
By investigating the first few loop orders, we observed that the c-discontinuity of the pentaladders is similarly very simple. Here we will derive such a simplification directly from the integral representation of section 3, where the discontinuity will collapse the ν integral onto a small circle as in the preceding example.
The c-discontinuity represents a cut along the channel carrying momentum along the ladder, from switching the sign of x 2 14 from spacelike to timelike. In terms of the cross ratios u, v, w in eq. (2.24), it can be implemented by Normalizing it so that ln c has discontinuity 1, the discontinuity is properly defined as To translate this to the x, y, z variables we use eqs. (3.5) and (3.6): After the continuation we have u, v < 0, which corresponds to x > 1 and y > 1, with x just below the x > 1 branch cut in the e 2πi c term, and similarly for y. Because both u and v rotate in the same way in eq. (5.18), we also need xy to acquire the phase 2πi. This phase can be put in either x or y, with z not rotating; the result will be physically equivalent due to the single-valuedness constraint enforced in section 3.2. We conclude that a valid continuation path for the e 2πi c term is to take x > 1 and y > 1 below the cut, but with a prior rotation of y around the origin. The discontinuity in the finite-coupling formula (3.26) for Ω is trickier to compute than that for the box ladder f because x > 1 ends up outside the radius of convergence of the hypergeometric series. We deal with this complication by using the standard hypergeometric transformation law under x → 1/x to express the result in terms of new functions with argument 1/x. Applied to eq. (3.20), this transformation yields hypergeometric functions with spin and dimension effectively interchanged: (Note that the transformation ν → ij, j → iν is equivalent to ν 2 → ν 2 + 4g 2 , g 2 → −g 2 in eq. (3.18), which is also equivalent to letting x → 1/x in the differential operator. See ref. [114] for further examples of such transformations.) The analytic continuation of F to x > 1 − i0 is then written as while for y we get an additional factor of e −πν , accounting for the rotation around the origin.
For the e −2πi c term the phases just get reversed, allowing us to compute the discontinuity: When we subtract the ν → −ν term in the integral (3.26), after noting thatF ν j | ν→−ν =F ν j , the trigonometric factors simplify dramatically and we end up with only: This result should be compared with eq. (5.16): the power laws for the box ladders have simply been replaced by hypergeometric functions. A further simplification, as in the box ladder case, is that the integral can be rewritten as a contour integral. Because the integrand is symmetrical in j = i ν 2 + 4g 2 ↔ −j, it does not have branch points at ν = ±2ig, but to discuss its terms separately we need to choose a branch. We pick the one with j ∼ iν at large |ν|. By the analysis leading to eq. (4.5), the functionF ν j(ν) (x) then only has poles in the lower-half-plane and is analytic in the upper halfplane (for x > 1). The functionF ν −j(ν) (x) has the opposite properties. Shifting the contour to an imaginary part just below −2ig, and closing the contour as in the box ladder case, we conclude that the c-discontinuity is saturated by the integral over the short cut from −2ig to 2ig of the first term: This formula is the main result of this subsection. An immediate consequence is that the dependence on z in perturbation theory occurs solely through powers of ln z.
More generally, at weak coupling, the integrand of eq. (5.25) can be series expanded in ν and j, which are uniformly small over the contour. Just as for the box ladder discontinuity (5.17), only odd powers of j contribute to the integral. It is helpful to explicitly pick out the odd part and divide it by j. At this point we also recall that the Ω pentaladders were identified in eq. (3.49) as one component of a four-vector. Repeating the calculation for the other integrals, we write where we have set s = iν/2 for future convenience. The result eq. (5.25) then implies that The other entries share the generic form 28) and are given explicitly as whereF ν j is the hypergeometric function entering the x → 1/x transform of eq. (3.29): We now describe this result explicitly at weak coupling.

Perturbative expansions and coproducts of c-discontinuities
The discontinuity integrand in eq. (5.26) can be doubly Taylor-expanded in s and g 2 : In general, we expect each term to involve powers of ln z, as mentioned, as well as polylogarithms of x and y originating from the expansion of the hypergeometric functions. For example, taking the g 2 → 0, s → 0 limit of eq. (5.28) using the methods of section 4, we get 2,0 = 1 2 ln(xy),Ṽ A good way to see these weights and to compute the higher-order terms is to use the differential equation that theṼ (L) i,k satisfy. Before taking the discontinuity, by using the properties of the hypergeometric functions we found a set of four coupled first-order equations, eqs. (3.50) and (3.51). Not surprisingly, since discontinuities and derivatives commute, Disc c V i (s, g 2 ) satisfies precisely the same coupled equations. Using eq. (5.3) we can rewrite the system as a coaction i,k functions are encoded in the powers of g and s. Namely, if we assign these expansion parameters both transcendental weight minus one, the diagonal terms in the matrix have weight zero, the upper-triangular entries have weight minus one, and the lower-triangular terms have weight one. For fixed L and k, coacting onṼ (L) j,k with M ij (s, g 2 ) will thus increase the weight of the iterated integrals in each entry by one, but the resulting entries should be interpreted as multiplying different powers of s and g in the expansion (5.32).
This coaction can be used to construct the functionsṼ (L) i,k iteratively, using a single boundary condition, which we will describe shortly. One begins with the transcendental weight 0 vector and coacts on it using the matrix M to get the complete weight-one component Reading off the second and third element, we reproduceṼ 2,0 = 1 2 ln(xy) andṼ 3,0 = − 1 2 ln c. The first element has a factor of s, so it corresponds to a term with k = 1, in particular V (1) 1,1 = ln z.
Following this procedure further, one can construct the c-discontinuity of our basis functions to any weight by iteratively coacting with matrix M . At each step one has to supplement the information from the coproduct with one boundary condition, because (being effectively a derivative operator) ∆ •,1 kills all constants. A convenient limit can be given at (x, y, z) = (1, 1, 1) (corresponding to (u, v, w) = (−∞, −∞, 1)). There we find from eq. (5.28) thatṼ 2 andṼ 4 vanish, while lim x,y,z→1Ṽ where j = 2 s 2 − g 2 as above, and γ E is the Euler-Mascheroni constant. ForṼ 3 we have dropped the singular logarithm ln c in this limit, in order to focus on the constant piece. Up to weight 4, these can be expanded explicitly as lim x,y,z→1 In general, we find that the top row and ratio of the first and third can be expanded into even and odd zeta values respectively: lim x,y,z→1Ṽ x,y,z→1Ṽ Referring back to eq. (5.33), we see that the functions there match eq. (5.41), modulo ln c terms, in the limit x, y, z → 1 without the addition of any constants, as needed (remember that the first entry in (5.38) should be compared to the g 0 s 1 term in (5.41)).
To getṼ (1) 4,0 and the weight-two contributions to the other functions, we now coact with M on the weight-one vector in (5.38), which we have promoted to a vector of full functions. This gives 2,1 = 1 2 ln z ln(xy) ,Ṽ The space of functions generated by this procedure turns out to be one we have already encountered-the space of SVHPLs introduced in eqns. (5.10) and (5.11), where z andz are now equal to 1 − x and 1 − y. For instance, we can rewrite all the functions we have computed above as 1,2 = L 1,1 + 1 2 ln 2 z ,Ṽ where we have left the SVHPL arguments {z,z} = {1 − x, 1 − y} implicit. We wish to show that the dependence on x and y for arbitrary weight is captured by SVHPLs with arguments 1 − x and 1 − y. Given the letters S disc in eq. (5.7), the main issue is to show that the functionsṼ (L) i,k (x, y, z) are single-valued at 0, 1, ∞ in the complex plane for (x, y). It is sufficient to look at two of the three limits, say where x and y both approach 1 or both approach ∞. We can probe the first limit with the boundary conditions (5.41), which tell us that the monodromies around this point are dictated by the first and third columns of the matrix M in eq. (5.37). The first and third columns contain only ln z, ln(xy), ln(x/y) and ln c = ln[(x − 1)(y − 1)]. The first three of these functions are smooth or vanish as (x, y) → (1, 1), and the fourth is real analytic (single valued). In other words, the potentially problematic entry ln[(x−1)/(y−1)] in the second and fourth columns is killed by the boundary condition (5.41). (This boundary condition is for z = 1, but the z dependence factorizes.) The single-valuedness at x, y → ∞ can be seen by considering eqns. (5.28), (5.29), and (5.30), as well as the expansions (5.21) and (5.31), whereby the dependence on x and y in Disc c V i (s, g 2 ) takes the form (xy) ±j times a regular expansion in powers of 1/x and 1/y in each term. Thus, the functions in Disc c V i (s, g 2 ) are single-valued in 1 − x and 1 − y, making them SVHPLs.
In an ancillary file, omegacdiscwt0-12.m, we provide the SVHPL representation of all the c-discontinuity functionsṼ (L) i,k (x, y, z) through weight 12. Finally, to return to the ladder integrals themselves, we insert the series expansion of Disc c V i (s, g 2 ) in eq. (5.32) and the series expansion of the square root in terms of Catalan numbers, i,k , as opposed to looking only at the combinations in eq. (5.48), is that this set is closed under the coaction. This allows theṼ (L) i,k to be computed recursively in an efficient manner, and will be critical to "undoing" the discontinuity.

The Ω-functions in the coproduct formalism
Having now constructed a basis of functions describing the c-discontinuities of the Ω system and their derivatives, our next task is to "undo" the discontinuity to get the full functions. As in the box ladder example, the key is that Steinmann hexagon functions without c-discontinuity are extremely constrained.
A function with no c-discontinuity must have first entries in {a, b}. Using eqs. (B.15) and (B.16) one can check that such functions can only have symbol letters in the set {a, b, m w }. Functions of this type were classified in ref. [69]; they are a subset of the functions called K functions there. These K functions can be expressed simply as products of logarithms ln k (a/b) and HPLs in m w . Thus, the c-discontinuity uniquely fixes the coproducts of our functions of interest, up to a few such K functions! Since all the non-log, harmonic-polylogarithmic behavior of these K functions depends on w, they are naturally probed by values of the i loop integral Ω (i) on the line where u = v = 1, namely Ω (i) (1, 1, w) as given in eq. (4.38). In fact, with a bit of trial and error, we find that a special combination always occurs, where x is the greatest integer less than or equal to x, and f 0 (u, v, w) = 1. Note that ln(u/v) = 1 2 ln(a/b). For the Ω space, we need only two additional transcendental K functions at each weight: These two functions can be constructed recursively from their nonvanishing coproducts: together with the boundary condition at (u, v, w) = (1, 1, 1): −Ω (k/2) (1, 1, 1) = (2k − 2)ζ k k even. i,k functions contain two types of terms. The first type involves other functions in V; they are determined by the matrix (5.37), which in the alphabet (5.5) reads (5.55) The second type of terms involves κ andκ, and we have determined them by solving integrability conditions. (By "integrability" here we refer to the commutativity of partial derivatives, not to be confused with quantum integrability.) Given the nonvanishing coproducts of these functions, one can define the Ω system recursively. The most complicated of these involve m u and m v for the even functions, and y u and y v for the odd functions: The first three terms on the first line of these, for example, come from the m u and m v terms in the first row of eq. (5.55). The remaining nonvanishing coproducts are then: There are a few exceptional cases at low weights: In other words, we can fix the boundary conditions for the coproduct description along the entire line (1, 1, w), not just at the point (1, 1, 1) given in eqs. (4.37), (4.39) and (4.48).
Finally, let us be explicit as to how the actual ladder integrals sit inside this basis, as Catalan-weighted sums along the same lines as eq. (5.48): , Ω Note thatΩ (1) e is not pure, so we should use eq. (3.30), not eq. (5.71) for that case. Also, the first two instances of W (L) and the first instance of Ω (L) are exceptional, needing additional κ contributions: ) Apart from these exceptions, eqs. (5.71) and (5.72) locate the five integrals per loop perfectly inside the Ω space for all L ≥ 1. The space of Ψ functions for the pentabox ladders has an analogous description, which is not surprising since the Ψ integral is obtained from the Ω integral by letting w → 0, Ψ (L) (u, v) = Ω (L) (u, v, 0). However, not all of the integrals are nonsingular in this limit. Instead of the (2n + 1)-dimensional space (5.53) at weight n, the following subspace survives,  We remark that the perturbative Ω space is much smaller at each weight than what would be obtained solely by imposing proper branch cuts (at weight one) and the constraints of the Steinmann relations at weight two. As we describe in appendix C, there are additional constraints on pairs of adjacent letters in the Ω space, that are reminiscent of the Steinmann relations. In appendix B, we mention that there are similar "extended Steinmann relations" that apply to the more general space of hexagon functions [74], and are related to the cluster adjacency principle [115].

5.6
Other Ω space properties and embedding into hexagon function space Although we have given a complete construction of the Ω space in the previous subsection, we can also ask how many functions can be obtained just as coproducts of a single function. This enumeration was useful for our initial understanding of the Ω space, before the above construction was discovered.
In particular, we can examine all the coproducts of the L loop odd ladder integral O (L) . Of the five ladder integrals at L loops, it has the highest weight, 2L + 1. We iteratively construct all of the {n, 1, 1, . . . , 1} coproducts of O (L) at weight n. These coproducts are highly degenerate, so we only keep the linearly independent span of them at each weight. Then we differentiate each of those functions to go to the next lower weight, and again keep only the linearly independent ones, and so on. The results for the dimensions of these spaces, and for just the parity-odd subspaces, are tabulated in table 2 for each L ≤ 6. Table 2 shows a few interesting properties. First of all, the dimensions are "palindromic": The number of independent functions increases by two with each successive differentiation,  Table 3. Dimension of the full weight-n Ω space, and decomposed into even and odd subspaces under parity P. In the P-even sector we also list the number of K functions, which have no y u , y v , y w letters in their symbol.
tracing the odd natural numbers, until it peaks and then declines again by two at each step, tracing out the same set of numbers. 8 The same palindromic property holds for just the subspace that is odd under parity P, although the peak position is shifted up in weight. Secondly, once the number of functions has reached its peak for a given L, the dimensions for weights below that peak equal the dimension of the full Ω space at that weight. We say that the space of coproducts becomes "saturated" below a given weight n e,o s (L), which depends on whether the parity is even (e) or odd (o). Higher loop orders do not give additional functions for weight n ≤ n e,o s (L), and the dimension n Ω (n) of the full Ω space can be read off. From table 2, we see that the dimensions saturate for even (or all) and odd functions at The total number of Ω functions at weight n is seen to be n Ω (n) = 2n + 1, matching the number in eq. (5.54).
In table 3 we list the dimensions of the even and odd subspaces for weight n ≤ 13. Parityodd functions necessarily contain the parity-odd letters y i ≡ {y u , y v , y w } in their symbols. The P-even subspace has a further subspace of "K" functions [69] whose symbols contain no parity-odd letters. These functions are simply HPLs with arguments 1 − 1/u, 1 − 1/v, and 1 − 1/w in some cases combined with logarithms. We have already identified two of them, κ n andκ n , but there are four more "secret" K functions in the Ω space, for a total of six at each weight n (for n > 3): Ω cyc , P odd 0 0 0 1 2 6 6 9 9 Table 4. Dimension of the full weight-n hexagon function space H, decomposed into even and odd subspaces under parity P, and compared with the corresponding dimensions for Ω cyc . In the P-even sector we divide the space into non-K and K functions.
In terms of HPLs, the four secret K functions at weight n are We have referred to the Ω space as a prototype or model for the full space of Steinmann hexagon functions H. How many of the functions in H does it capture or miss, as we go up in weight?
The Ω space has a particular orientation, while H is closed under all permutations of (u, v, w). We define Ω cyc to also include cyclic permutations of the Ω space functions under (u, v, w) → (v, w, u) and (u, v, w) → (w, u, v). For the most part, these permuted functions are independent. However, the top line of eq. (5.80) has two K functions, which after including cyclic permutations, become only three K functions in all, H n (1 − 1/u i ), i = 1, 2, 3. So we lose three K functions at each weight, and the number of K functions in Ω cyc is 3 × 6 − 3 = 15 at weight 6 and above. There is a similar degeneracy under cyclic permutation for the few parity-odd functions at weights 3 and 4, and for the non-K parity-even functions at weights 4 and 5. (In the latter case, certain linear combinations of cyclic permutations of non-K functions are actually K functions.) Beyond weight 5 there are no non-K degeneracies, and so the dimension of the weight n part of Ω cyc is 3(2n + 1) − 3 = 6n for n ≥ 6. This dimension grows only linearly with n, whereas the dimension of the weight n part of N grows much faster, roughly like 1.7 n .
In table 4 we compare the dimensions of the full hexagon function space H, which has been trimmed to remove all inessential functions [74], with the dimensions of Ω cyc through weight 8. We have split the functions into P even and P odd, and we have further split the P-even functions into K functions and non-K functions. The space Ω cyc spans the full hexagon function space through weight 3. At weight 4 it only misses 4 K functions, one of which is the constant ζ 4 . At weight 5 it misses numerous K functions, and a single P-even, non-K function, but it still captures all the P-odd functions at weight 5. The single missing P-even, non-K function evaluates to 5ζ 5 −2ζ 2 ζ 3 at (u, v, w) = (1, 1, 1), while the weight 5 part of Ω cyc vanishes at (1, 1, 1). Presumably this weight 5 function is a W-like seed for non-ladder DCI integral topologies beginning at three loops, in which three pentagons with appropriate numerators are joined together at a common vertex.

A nonperturbative coaction
That the dimensions of the Ω space saturate has an interesting implication: it allows us to define the coaction nonperturbatively. This can be illustrated by returning to the discontinuity functions Disc c V i (s, g 2 ), defined nonperturbatively in eq. (5.28). Using the differential equation these functions satisfy, we see they can also be defined by a path-ordered exponential, where the argument of the exponential is a 4 × 4 matrix: (5.81) Dotting U ij on the left with the initial condition in eq. (5.41) reproduces the vector Disc c V i , but this matrix contains additional transcendental functions. Roughly speaking, we expect this space of functions to be necessary and sufficient to describe all the possible analytic continuations of theṼ i . A nonperturbative coaction can then be defined simply as a matrix product: ∆U ij (s, g 2 ; x, y, z) = k U ik (s, g 2 ; x, y, z) ⊗ U kj (s, g 2 ; x, y, z) . (See refs. [116,117] for related constructions also involving hypergeometric functions.) The ∆ •,1 coproduct component from eq. (5.82) is easily seen to reproduce eq. (5.36), and perturbatively the ∆ 1,...,1 components reproduce the symbol of these functions. However, this equation defines ∆ nonperturbatively, and in particular for all ∆ •,k and ∆ k,• for k ≥ 1. (We haven't discussed boundary conditions, and in principle the definition in eq. (5.81) might need to be "twisted" by ζ-valued constants so as to match the ordinary coproduct of polylogarithms. We leave this exploration to future work.) Note that, by construction, the coaction (5.82) satisfies a coaction principle [87][88][89]. That is, the first entry of the coaction is always contained within the original space of functions. A similar (perturbative) coaction principle has been observed in the full space of Steinmann hexagon functions, where it constrains the transcendental constants that can appear in the first entry in addition to restricting the symbols of these objects [69,88]. The consequences of this coaction principle thus extend beyond what is currently understood in terms of physical principles (since only the symbol-level constraints are understood in terms of allowed branch cuts of Feynman integrals), as will be described in more detail in a forthcoming work [74].
The first entry of the coaction (5.82) manifestly realizes this same property. In particular, the first entry of the coaction maps to the same space of functions for any initial conditions one dots into U ij (after which the coaction principle more closely resembles those discussed in [88], since the second entry of the coaction will in general map to a larger space).
The nonperturbative coaction for the Ω system can be defined in an analogous fashion, writing in matrix form the general solution to the differential equations following from eqs. (5.56)-(5.69). The appearance of the κ andκ functions in this space implies that U will now appear as a 4 × 4 sub-block of a bigger matrix.
It is remarkable that the set of all coproducts of the Ω integrals (loosely speaking, the space of all their possible derivatives and analytic continuations) can be encoded nonperturbatively in a single matrix.

Conclusions
We have investigated a class of integrals, Ω (L) andΩ (L) , that have representatives at each loop order. In doing so, we have found something remarkable: that their all-orders behavior can be expressed in terms of beautifully simple integral formulas, given in eqs. (3.26) and (3.32). Using these expressions, we can extract any desired loop coefficient, and have control over the full behavior of the functions via infinite sums.
We have also investigated the coproducts of these functions to all orders, allowing us to characterize the complete space of polylogarithms that envelops the Ω (L) andΩ (L) integrals and their derivatives. As a consequence, we can now efficiently construct a subspace of the Steinmann hexagon functions to arbitrarily high weight. This space is equipped with a coaction both perturbatively and at finite coupling, and obeys a coaction principle.
The inevitable next question is, can we characterize the full Steinmann hexagon space H in a similar way? For example, can we find a systematic definition of the hexagon function coproducts, analogous to eqs. (5.56)-(5.64), which solves the integrability conditions to all orders? Or perhaps are there other subspaces of H, larger than Ω, that we can describe to all orders, that capture a wider set of functions that are not in Ω cyc ? Does the amplitude itself have a form like the Ω (L) andΩ (L) integrals, and could it be written as a finite-coupling expression involving (several) Mellin integrals? We suspect that this might be possible, although presumably it will have to include the full flux-tube dispersion relations [43,49].
One reason for suspecting this is that the radius of convergence of the perturbative expansion of the Ω integrals appears to be much larger than that for amplitudes, as discussed in section 4.4. The radius of convergence for amplitudes is relatively close to that for the cusp anomalous dimension, which also controls the behavior of the flux-tube expansion at finite excitation number. Thus it seems likely that the large-order behavior of six-point amplitudes is controlled by other families of integrals that grow more quickly than the ladders.
In the past, several of the authors have observed differential equations linking the amplitude at different loop orders [69,73]. While some of these relations do not hold at higher orders [74] they still suggest that a larger piece of the Steinmann hexagon space has an iterative or recursive structure which awaits exploitation. The representations for v, 1 − v, y v , and so on, can be obtained by cycling Z i → Z i+1 , remembering that under this transformation u → v → w → u, while y u → 1/y v → y w → 1/y u . As discussed in section 3, for many purposes, a better set of variables for the Ω integrals is {x, y, z}, where The inverse relations are, Notice that x and y depend only on y u y v and y w . That is, the only dependence on y u /y v is through z. Parity sends the dual coordinates x i → x i+3 (mod 6) and the momentum twistors Z i → Z i+3 (mod 6). Parity does not affect the cross ratios u, v, w but it exchanges √ ∆ ↔ − √ ∆, so that z + ↔ z − and y u , y v , y w are inverted: y i ↔ 1/y i . Eqs. (A.7)-(A.10) show that parity exchanges x and y, leaving z invariant.
The u derivative of a function F , holding v, w fixed, is given in terms of first coproducts by and derivatives with respect to v and w are obtained by cyclic permutations of this equation. Derivatives with respect to x, y, z are related to these derivatives by, (A.14) In discussing the Ω space of functions in section 5, it is very useful to change the hexagon alphabet from S hex in eq. (A.2) to Given coproducts labelled using S hex , we can convert them to those using S hex , by plus the relations obtained by cyclic permutations. To go in the opposite direction, we use, plus the cyclic relations. For example, the z derivative, using eq. (A.14) and expressed in terms of coproducts using the alphabet S hex , is The x and y derivatives are more complicated, but are a bit more simply expressed in terms of the y i variables and using the following combinations with opposite parity:

B Extended Steinmann Relations in the Full Hexagon Function Space
In this appendix we discuss properties of adjacent symbol entries in the full hexagon function space H, as a prelude to a similar discussion for the Ω space and its c-discontinuity in the following appendix.
One advantage of the alphabet S hex is that the Steinmann relations are made transparent, insofar as each letter a, b, c contains a unique three-particle invariant [ A similar simplification occurs in the heptagon letters used in [63,75], where each threeparticle invariant only appears in a single letter a 1j . Thus the Steinmann constraints [69], and permutations thereof, are solved (at symbol level) simply by requiring that the first two entries of the symbol do not contain any of the six combinations, However, by examining the double coproducts of functions obtained by taking multiple coproducts of high loop six-point amplitudes, we have found that the same constraint also holds deeper into the symbol. That is, the combinations never appear [74]. We refer to this condition as the extended Steinmann constraints. There are also 26 independent constraints on double coproducts from function-level integrability. Expressed in the alphabet S hex , they are contained in the following, When we combine the constraint (B.4) with the integrability constraints, we find 52 independent pairs of double coproducts. However, when we construct the space of (extended) Steinmann hexagon functions iteratively in the weight, imposing correct branch cuts along with the additional constraint (B.4), we find only 40 independent pairs [74]. Of these pairs, 24 are parity even and 16 parity odd. Interestingly, these pairs also match those provided by the "cluster adjacency" principle described in ref. [115], once integrability is imposed.
In order to show linear combinations of symbol entry pairs more clearly, we denote a pair of allowed final entries using the following notation (not to be confused with a similar notation used in superscripts in eqs. (B.5)-(B.11)): To denote cyclic classes, we write a i ∈ {a, b, c}, m i ∈ {m u , m v , m w }, and y i ∈ {y u , y v , y w }. Again, i = j = k. In this notation, the 16 odd pairs are The spaces of single and double coproducts are much smaller in the Ω subspace than in the full space of Steinmann hexagon functions described in appendix B. At the single coproduct level, parity even functions in the Ω space are observed to have only 8, not 9 final entries: y u and y v do not appear separately, but only the combination y u y v . That is, E yu − E yv = 0

C.2 Coproduct relations for the c Discontinuity
We define the space Ω c to be the discontinuity in the c variable of all the functions in the Ω space.
In Ω c , the set of allowed single and double coproducts shrinks even further. Also, one of the three derivatives simplifies considerably. At the same time, we lose almost no information, because only the functions κ andκ are set to zero, while the remaining functions are still linearly independent.
In particular, there are only five letters in the alphabet for Ω c : {c, m u , m v , y u y v , y w }. In this subsection we provide coproduct relations between the integrals Ω (L) ,Ω (L) and O (L) . While it is possible to read off all such relations from results in section 5.5, we can also derive many of them directly from the differential equations they satisfy. These relations were useful when constructing Ω (L) at higher loops in an earlier stage of this work, and they serve to illustrate the structure of the Ω functions in the coproduct formalism. The relations are all valid at sufficiently high loop order, using starting at either L = 2 or 3.
In appendix A, the x, y, z derivatives of any functions F are expressed in terms of coproducts for the alphabet S hex . Consider, for example, the operator z∂ z . Its action can be written as When applying a second-order differential operator, such as those appearing in the differential equations for the weight-2L transcendental function Ω (L) , the weight can be reduced either by one or two. The former case occurs when the second derivative hits the rational factor in eq. (C.9) instead of the transcendental function. In the case of eq. (3.14), using the analogous expression for x∂ x , we find that these weight-(2L − 1) terms combine to Since the action of the operator should give Ω (L−1) , which has uniform weight 2L−2, the righthand side should vanish. This condition, together with the parity conjugate relation (3.15), implies that Ω a + Ω b − Ω c + Ω mw = 0, Ω yw = 0. Now we look at derivatives of O. We start with the relationΩ o = −z∂ z O and set F = O in eq. (C.9). BecauseΩ o is a pure transcendental function with no rational prefactor, all the terms containing non-constant rational prefactors must vanish. It is easy to see that no linear combinations of F a , F b , F mw and F yu − F yv can produce a constant prefactor. Hence we obtain, (C.14) Next we insert eq. (C.14) into eq. (A.21) for (x∂ x + y∂ y )O, which appears in eq. (3.39) for Ω: In summary, since we know all three derivatives of the odd ladder O, we can determine all nine of its first coproducts, Returning to the derivatives of Ω, we find empirically that z∂ z Ω is a pure function. This fact implies, via eq. (C.9), that Ω a = Ω b = Ω c = 0. (C.20) These additional relations imply that x∂ x Ω = 1 2 (Ω mu + Ω mv ) − Ω yu , y∂ y Ω = 1 2 (Ω mu + Ω mv ) + Ω yu , (C.21) z∂ z Ω = 1 2 (−Ω mu + Ω mv ) .

(C.22)
We have also found some first-order coproduct relations forΩ: These relations are equivalent to is indeed the same as the one appearing in the z derivative (C.26). One can also show to all orders that (x∂ x + y∂ y )Ω = y 1 − y (x∂ x + z∂ z )Ω + x 1 − x (y∂ y − z∂ z )Ω . (C.28) Inserting the coproduct representations of these derivatives, given above, we find relations between the first coproducts ofΩ and Ω: −Ω c +Ω yw = Ω mu + Ω yu ,Ω c +Ω yw = −Ω mv + Ω yu .

C.4 Improving the MHV-NMHV operator
Through five loops, the six-point MHV amplitude E and NMHV amplitude E obey a curious relation that connects these amplitudes at different loop orders [69]. If we perform a cyclic permutation u → v → w → u on that relation, in order to give it the same u ↔ v symmetry as the Ω integral, it becomes X old [E(u, v, w)] = g 2 (2E(v, w, u) − E(u, v, w)), (C.32) where X old [F ] ≡ −F w,w − F 1−w,w − 3F yw,yw + F yu,yu + F yv,yv + 2(F yu,yw + F yv,yw ) − F yu,yv − F yv,yu (C.33) is written in terms of the old alphabet S hex . (For an earlier version of this relation, see ref. [73].) In fact, this relation fails for amplitudes at six loops [74]. However, we will see that a version of it survives to arbitrary loop order in the pentaladder integrals.
In general, E obeys many relations on its double coproducts, which allow the operator X to be rewritten without changing its action on E. However, its action on the ladder integrals will generically change. It turns out that a better form for X, written in terms of the new alphabet S hex , is X[F ] = −3F yw,yw + F yu,yu + F yv,yv + 2(F yw,yu + F yw,yv ) − F yu,yv − F yv,yu + F a,a + F b,b + F c,c + F a,mw + F b,mw − F mw,a − F mw,b − 2(F c,mu + F c,mv − F mu,c − F mv,c ) .
(C. 34) This form is better because X now has a very simple action on the ladder integrals. We find that , and X[W(w, u, v)] at both L = 2 and L = 3. But above three loops, there are no anomalies in the action on these integrals to any order. This can be contrasted with the operator's action on E, which remains anomalous at six loops. The operator X also has an interesting action on the full Ω space. In particular, note that for each ladder integral considered above (and ignoring low-weight anomalies), X[F (v, w, u)] = X[F (w, u, v)] = 0. While this is not quite true for the full space, we do find that for a general function F (u, v, w) ∈ Ω, X[F (v, w, u)] , X[F (w, u, v)] ∈ {κ,κ} . (C.39) That is, the action of the operator X on cyclic rotations of functions in the Ω space can be expressed entirely in terms of κ andκ functions of the appropriate weight, which vanish on the c-discontinuity. In effect, the operator X annihilates the c-discontinuity of the cyclic and anti-cyclic rotations of the Ω functions. This is a surprising property, and one that suggests further investigation.