The Differential of All Two-Loop MHV Amplitudes in N=4 Yang-Mills Theory

We present an explicit analytic calculation of the differential of the planar n-particle, two-loop MHV scattering amplitude in N=4 super Yang-Mills theory. The result is expressed only in terms of the polylogarithm functions Li_k(-x), for k=1,2,3, with arguments x belonging to the special class of dual conformal cross-ratios known as cluster X-coordinates. The surprising fact that these amplitudes may be expressed in this way provides a striking example of the manner in which the cluster structure on the kinematic configuration space underlies the structure of amplitudes in SYM theory.


Introduction
Insights gleaned from difficult calculations have driven much of the recent progress in unlocking the structure of N = 4 super Yang-Mills (SYM) theory. The synergy between technical and conceptual advances frequently enables the undertaking of previously impossible calculations; the often surprising results of which in turn fuel a deeper understanding of the theory. Indeed, in SYM theory the search for such insights, rather than the need for busywork, is precisely the motivation for carrying out arduous new calculations.
In this note we complete the analytic calculation of the differential dR (2) n of the planar two-loop n-particle MHV amplitudes (or, more properly, the 'remainder functions'with infrared divergences subtracted off in a standard way [1,2]). Reasonably efficient numerical techniques for evaluating R (2) n have been available for some time [3], and analytic formulas are known in two-dimensional kinematics [4] but a breakthrough towards unlocking the general analytic structure of these amplitudes was made by Caron-Huot in [5]. By considering the dual superconformal symmetry of a certain generalization of scattering amplitudes depending on twice the usual number of Grassmann variables, he was able to express dR (2) n in terms of a certain combination of one-fold integrals. Here we complete the evaluation of these integrals and present analytic expressions for dR (2) n . We find that they can be expressed completely in terms of the functions Li k (−x), for k = 1, 2, 3 with arguments x always belonging to the set of cluster X -coordinates on the kinematic configuration space Conf n (P 3 ). This provides strong support for the suggestion of [7] that the cluster structure of this space underlies the structure of amplitudes in SYM theory.
While this computation may seem modest in scope, we feel that it is a useful example in which to showcase the power of the two most recent additions to the amplitudeologist's toolkit: positivity and cluster X -coordinates. In particular, the positive domain, a subset of the kinematic domain Conf n (P 3 ), is evidently the natural habitat for amplitudes in SYM theory, and cluster X -coordinates provide a natural set of arguments for these amplitudes to 'depend on'. It is our hope that other analytic formulae in SYM theory can be unlocked using these or similar approaches. In particular, it would be very interesting to see if the monodromies of R (2) n , which were computed in integral form in [8] (see also [9]), can be similarly expressed only in terms of Li k (−x).
Following ample motivation (but minimal review), we present our analytic expression for dR (2) n . For additional introductory material, we refer the reader to [7] and the references therein.

Motivation and Review
In [5], Caron-Huot expressed the differential of the planar two-loop n-particle MHV amplitude in terms of momentum twistors [6] as and presented a means of calculating the coefficient functions C i,j . Already the fact that the differential may be expressed in the form of eq. (2.1) is a nontrivial all-loop-order prediction of [5].
Thanks to the full dihedral symmetry of the amplitude in the particle labels, it is sufficient to focus on C 2,i . This was given as a sum of four components C (a) 2,i , for a = 1, 2, 3, 4, two of which were represented analytically as where for our purposes it is sufficient to take x 2 i,j = i i+1 j j+1 , and where u i,j,k,l = i i+1 j j+1 k k+1 l l+1 i i+1 k k+1 j j+1 l l+1 (2.4) and u ≡ u 2,i−1,i,1 . C 2,i has the integral representation and C 2,i may be determined from C (2) 2,i by imposing dihedral symmetry on eq. (2.1), which fixes C 3) along with the integral representation in eq. (2.5) were sufficient to compute the symbol of dR (2) n in [5], any attempt at an analytic expression for dR (2) n would have encountered two major obstacles:

Choosing a 'Good' Kinematic Domain
First of all, n-particle scattering amplitudes in SYM theory are multi-valued functions on the 3(n − 5)-dimensional kinematic configuration space Conf n (P 3 ) (we refer the reader to [7] for a review of this notation), and in computing scattering amplitudes it is often very difficult to get all of the branch cuts in the right place. In particular, Mathematica's Integrate[] function has no a priori understanding of Conf n (P 3 ) and cannot be expected to easily return an analytic formula valid on the whole of this domain-at best it can produce numerical results at specifically inputted kinematic points, or an analytic formula valid on some subdomain of its own choosing. On physical grounds, amplitudes should be real-valued and singularity-free throughout the Euclidean domain, a subset of Conf n (P 3 ). Even for what is in some sense the simplest non-trivial multi-loop amplitude, R 6 , finding a formula with these properties was by far the hardest part of [10]. For n > 6, where the kinematic configuration space is far more complicated (due to Gram determinant constraints amongst dual conformal cross-ratios), finding explicit formulas valid throughout even just the Euclidean domain seems like a daunting challenge. Instead of giving up all hope, one would be content to find expressions valid even in some subset of the Euclidean domain. But, which subset should one look at? What principle, either mathematical or physical, could make any one subset more worthy of attention than another?
The second complication has to do with the class of iterated integrals of the type which define generalized polylogarithm functions and which appear in the result of [5]. Such functions can be partially characterized by their symbols, but integrating a function of this type can generate a function whose symbol contains entries that are, in general, arbitrary alebgraic functions of the entries in the symbol of the original function. Already Caron-Huot's result for the symbol of R (2) n contains rather non-trivial algebraic functions on Conf n (P 3 ), and one might have worried that even more complicated functions could appear after integration. Yet, true believers in SYM theory know well that it only ever produces very special polylogarithmic functions, not 'general' ones. One would therefore have every reason to expect only relatively straightforward expressions to actually appear in amplitudes. Unfortunately, unless one knows in advance what to look out for, the plethora of identities amongst polylogarithm functions makes it difficult to identify any particular set of arguments as preferred over any other set.

Tools and Techniques
Fortunately, related recent advances have shown how to overcome both obstacles at the same time.

Cluster X -coordinates
We begin by addressing the question of Li k arguments by noting that in [7] it was shown via examples at n = 6, 7, and suggested more generally, that (in a sense made precise in that paper) two-loop MHV amplitudes only 'depend on' certain very special variables. These variables are called cluster X -coordinates on the kinematic configuration space Conf n (P 3 ), which may be realized as the Grassmannian quotient Gr(4, n)/(C * ) n−1 . For the purpose of this paper, the notion of 'depending on' only certain variables can be stated very clearly: in support of [7], we find that the differential dR (2) n of all two-loop MHV amplitudes can be expressed only in terms of the functions Li k (−x), for k = 1, 2, 3 with arguments x always belonging to the set of cluster X -coordinates on Conf n (P 3 ).
In fact, we find that (in line with the expectation expressed in the previous section) only relatively tame X -coordinates appear in dR (2) n , for any n. The most complicated of these is the cross-ratio which depends on the eight points {1, 2, 3, j − 1, j, j + 1, i − 1, i}. Here we use the notation ab(cde) ∩ (f gh) ≡ acde bf gh − bcde af gh .
The fact that the complexity of two-loop MHV amplitudes stabilizes (in this sense) already at n = 8 is manifest in the result of [5]. This implies that a full understanding of the cluster structure on Conf 8 (P 3 ) should be sufficient to understand the structure of all such amplitudes.

The Positive Domain
We now turn to the question of the kinematical domain, where the sole dependence on cluster X -coordinates plays a fortuitous role. A salient feature of cluster X -coordinates is that they are positive-valued everywhere in the positive domain, which is the subset of the Euclidean domain defined by imposing that abcd > 0 whenever a < b < c < d.
With the differential dR (2) n expressed completely in terms of the functions Li k (−x), it therefore becomes manifestly real-valued and singularity free throughout the positive domain. (Moreover, we believe that R (2) n itself is also positive-valued throughout the positive domain, but that is another matter.) The importance of positivity for scattering amplitudes was first emphasized in [11], though in a somewhat different context. Their Grassmannian for the integrand involves both 'external' kinematic data as well as 'internal' loop integration variables, but it is now clear that positivity in the 'external' data alone plays a similarly important role for the fully integrated amplitudes which we study here.
Before proceeding let us add one crucial disclaimer. As reviewed in [7], for n > 7 the cluster algebra associated to Conf n (P 3 ) has an infinite number of cluster X -coordinates (although of course only a finite number actually appear in dR (2) n ). Moreover there is no known general classification of these coordinates, and given any cross-ratio there does not exist a general algorithm for determining whether or not it actually is a cluster X -coordinate. Therefore, for n > 7 we use the empirical criterion discussed in section 6.6 of [7]: we say that a cross-ratio x is a cluster X -coordinate if 1 + x factors into a product of four-brackets as a consequence of Plücker relations and if x > 0 everywhere inside the positive domain.
Given the two advances we have reviewed-the understanding that the natural domain on which to study multi-loop amplitudes in SYM theory is the positive domain, and that the natural set of arguments appearing inside the Li k functions for two-loop MHV amplitudes are cluster X -coordinates-analytically integrating eq. (2.5) transforms from being merely possible to being inevitable.

Results
In this section we present the Li 3 and Li 2 × Li 1 contributions to dR (2) n . There are, in addition, terms of the form Li 3 1 and π 2 Li 1 which are too numerous to efficiently display here. For this reason we attach to this submission a Mathematica notebook which contains the necessary expressions and which can construct the full analytic formula for dR (2) n for any given n. While the original integral representation of C i,j required the general object u i,j,k,l , the integrated result can be written in terms of the slightly smaller class of cross-ratios defined by u i,j,k = u i,j,k,i−1 . The cluster X -coordinates related to the three-index u's (in the same sense that the v i and the u i used in [7] for n = 6 are related to each other) Here we use the notation a(bc)(de)(f g) ≡ abde acf g − abf g acde . For any n, v i,j,k is a cluster X -coordinate as long as i < j < k (mod n). We also find it useful to define another type of ratio, which is a cluster X -coordinate when i = j ± 1. Because we are interested only in objects of the form Li(−x), where x is a cluster X -coordinate, we will make one more definition li k (x) = Li k (−x) (4.4) purely in the interest of cleaning up the notation. We begin with the essentially semantic task of converting C (1) and C (4) into Xcoordinate form. Expressing C (4) 2,i in the desired form is trivial: C (1) is slightly more difficult, since it involves non-conformal objects such as log Of course, the expression as a whole is conformally invariant, but it takes some combinatoric effort to recast things as an explicit sum over conformal cross-ratios. This is only a problem for the O(Li 3 1 ) ratios, our results for this are in the attached Mathematica notebook. For Li 2 × Li 1 , we find that the ratios appearing are in fact already in cluster X -coordinate form, giving This sum looks slightly different than eq. (2.2) because we find it useful to make explicit the behavior of 1 − u j,k,k−1,j+1 at various upper and lower limits of the j, k summation indices. The double-sum of eq. (4.6) contains some boundary terms which divergespecifically, when j, k equal 2, n or i − 2, i. However, these divergences cancel when the O(Li 3 1 ) terms are added. The attached Mathematica file identifies and discards these canceling divergent terms.
We now turn to the main focus of this note: the evaluation of the remaining integrals in C (2) 2,i . For fixed i, j, the terms on the first four lines of eq. (2.5) altogether depend on at most nine distinct momentum twistors. We can therefore focus our efforts entirely on evaluating the integrals for the case n = 9, i = 8, j = 5, and then replace {4, 5, 6, 7, 8, 9} → {j − 1, j, j + 1, i − 1, i, i + 1} to recover the summand for arbitrary i, j. One may immediately worry about divergences at boundary terms in the sum, such as j = i − 2; while these cases do create individual terms that are divergent, these divergences cancel out in the full function.
In order to present our result for C 2,i in a succinct fashion, let us define the following permutation operators acting on the particle labels: as well as the four cross-ratios . (4.10) These ratios have the following 1 + R i factorizations: None of the R i depend on more than eight points, and we have checked that they are all X -coordinates of the Gr (4,8) cluster algebra (for more information on generating cluster X -coordinates via mutation, see [7]).
Note that there is additional symmetry present in in the arguments of R 1 and R 2 , which are related via R → 1/R combined with i ↔ j.
Given these definitions, we find that C 2,i has the following relatively simple Li 3 contribution: The three symmetries lead to 32 distinct Li 3 's in each term in the j sum. This presentation has discarded all terms which cancel telescopically in the sum. One can alternatively incorporate the non-summand terms into the summand by including some telescopic cancellations, thus increasing the total number of Li 3 terms (still all with cluster X -coordinates as arguments) in the summand to 48.
While there are no remaining telescopic cancellations in eq (4.12), the full 32 Li 3 terms do not appear for all values of i and j 1 . For example, six of the Li 3 terms go to zero at the upper limit of the sum, j = i − 2.
It is interesting to note that the evaluation of the integrals in eq. (2.5), for the general case n = 9, i = 8, j = 5, necessarily produces Li 3 's with non-cluster arguments. However, these terms cancel telescopically in the j sum and are zero at the j = 4 and j = i − 2 boundaries.
Next we turn to the Li 2 × Li 1 contribution to C 2,i , which we represent here as (4.13) Note that σ a v a,b,c is not a cluster X -coordinate, so some of the terms in eq. (4.13) aren't obviously in the form li k (x). However, this obstruction is easily removed by the relationship σ a li 1 (v a,b,c ) = − li 1 (v a,b,c ).
We have checked our full result for C 2,i against numerical integrations of (2.5) for several hundred random kinematic points in the positive Grassmannian for n ≤ 12. 1 The exact counting is: for n > 6 the total number of Li 3 terms in C (2) 2,i is given by 32i − 169 for 5 < i < n and 26n − 141 for i = n. For n = 6, C 2,6 has 11 Li 3 terms. C It is important to emphasize that our particular expression for eqs. (4.13) is far from being unique. Some additional symmetry can be introduced by adding terms that telescopically cancel, as was the case with the Li 3 sum. Furthermore, there exist numerous non-trivial identities amongst the Li 2 terms in eq. (4.13), introducing additional redundancy. This particular representation was chosen simply because it was the most typographically concise presentation we could deduce.
We conclude this note with a brief discussion of parity (see Appendix A of [7] for a thorough introduction). The parity invariance of dR (2) n follows from the fact that * C i,j = C j,i and i C i,j = 0. These properties can be made manifest at the level of the symbol, as was done in [5], but our integrated results sacrifice explicit parity invariance for brevity. Of course, parity invariance in our functional representation can be confirmed through the application of (numerous) polylogarithm identities.