4-particle amplituhedron at 3-loop and its Mondrian diagrammatic implication

This article provides a direct calculation of the 4-particle amplituhedron at 3-loop order, by introducing a set of practical tricks. After delicately rearranging each piece of this calculation, we find a suggestive connection between positivity conditions and Mondrian diagrams, which will be quantitatively defined. Such a pattern can be generalized for all Mondrian diagrams among all those contribute to the 4-particle integrand of planar N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} SYM to all loop orders, as a subsequent work 1712.09994 will show.


Introduction
The amplituhedron proposal for 4-particle integrand of planar N = 4 SYM to all loop orders [1,2,3,4] is a novel reformulation which only uses positivity conditions for all physical poles to construct the loop integrand. At 2-loop order, as the first nontrivial case, we have just one (mutual) positivity condition D 12 ≡ (x 2 − x 1 )(z 1 − z 2 ) + (y 2 − y 1 )(w 1 − w 2 ) > 0, (1.1) where are all possible physical poles in terms of momentum twistor contractions, and x i , y i , z i , w i are trivially set to be positive for the i-th loop. The resulting integrand is the double-box topology of two possible orientations, and it is symmetrized for two sets of loop variables [2]. As the loop order increases, its calculational complexity grows explosively due to the highly nontrivial intertwining of all L(L−1)/2 positivity conditions of D ij 's. As far as the 3-loop case, it is done under significant simplification brought by double cuts [2], still there is considerable complexity that obscures its somehow simple mathematical structure, as we will reveal in this article and the subsequent work [5].
As an illuminating appetizer, we reformulate the 2-loop case in the following. As usual, let's preserve z 1 , z 2 for imposing D 12 > 0, and triangulate the space spanned by x 1 , x 2 , y 1 , y 2 , w 1 , w 2 . We introduce the ordered subspaces characterized by, for instance: which is a d log form (omitting the measure factor) of the orderings x 1 < x 2 , y 1 < y 2 and w 1 < w 2 . In this particular subspace, positivity condition (1.1) unambiguously demands where x 21 ≡ x 2 − x 1 and so forth. Here, x 21 , y 21 , w 21 can be treated as genuinely positive variables which replace the original x 2 , y 2 , w 2 . Then the relevant d log form for z 1 , z 2 is simply 1 z 2 (z 1 − z 2 − y 21 w 21 /x 21 ) = x 21 z 1 z 1 z 2 D 12 , (1.4) analogously, for X(12)Y (12)W (21) we have x 21 z 1 + y 21 w 12 z 1 z 2 D 12 . (1.5) A seemingly farfetched observation is, after we flip W (12) to W (21), the additional term y 21 w 12 appears in the numerator above due to the orderings of y 1 , y 2 and w 1 , w 2 are now opposite, allowing one to orient the double box "vertically", as explained diagrammatically below.
In figure 1, we have chosen two perpendicular directions for x and y, while the z and w directions are opposite to those of x and y respectively. Then we assign each loop with a number as usual, but now these numbers have a meaning of orderings of positive variables. Since loop number 2 is below 1, we naturally interpret this as y 2 > y 1 , and similarly w 1 > w 2 . In this way, it is straightforward to conclude that, if we flip w 1 > w 2 back to w 2 > w 1 , there is no consistent way to place loop numbers 1, 2 vertically so the double box can be only oriented horizontally! After we sum the numerators above over W (12) and W (21) respectively, namely 1 w 1 (w 2 − w 1 ) (x 21 z 1 ) + 1 w 2 (w 1 − w 2 ) (x 21 z 1 + y 21 w 12 ) = x 21 z 1 + y 21 w 1 w 1 w 2 , (1.6)

Fundamentals of Positive d log Forms
First, we will extend the fundamentals of positive d log forms in [2], as the minimal techniques necessary for the posterior sections. It is known that, for a generic positive variable ranging from a to b (a < b), its d log form is given by finally, if b = a for the two special cases above, we have which will be named as the completeness relation. It has a natural interpretation as the sum of projective lengths of two complementary positive intervals. Note that we have treated a as a constant above, while it could also be a positive variable. In that case we only need an additional form 1/a, so the completeness relation now becomes 1 where the LHS characterizes nothing but two ordered subspaces in which x > a and x < a respectively. A trivial generalization of (2.4) for n x i 's satisfying x 1 . . . x n ≷ a is then here, for example, x 1 . . . x n > a is characterized by Another less straightforward generalization of (2.4) for x 1 +. . .+x n ≷ a is where both parts of the LHS can be proved recursively. If we assume they hold for x 1 +. . .+x n−1 > a and x 1 +. . .+x n−1 < a respectively, to obtain the form of x 1 +. . .+x n > a we must separate it into two parts as where in the first line, x n is positive in the first term, but greater than (a−x 1 −. . .−x n−1 ) in the second. The sum in the second line nicely returns to the form for n x i 's. To obtain the form of x 1 +. . .+x n < a, we can simply insert into the form for (n−1) x i 's, note that (a−x 1 −. . .−x n−1 ) is treated as one positive variable above. These two forms, as well as completeness relation (2.8), are often used in the subsequent derivation.
It is also convenient to introduce the co-positive product of forms. For example, for y > x 1 , . . . , x n , to obtain its form we can divide it into n! parts with respect to n! ordered subspaces in which x σ 1 < . . . < x σn and {σ 1 , . . . , σ n } is a permutation of {1, . . . , n}. Then we need to simplify by induction, here X n = {x 1 , . . . , x n }. Now let's focus on x n 's location in each permutation while omitting those of x 1 , . . . , x n−1 , it is straightforward to regroup the sum in order to reach (2.12) therefore here the symbol ∩ is the co-positive product operation. This product denotes the intersected subspace of a number of different subspaces as one form. If we evaluate the residue of I n at y = ∞, it returns to which is the completeness relation of n positive variables as all x i 's are trivially less than infinity.
Analogously, for y < x 1 , . . . , x n we need to simplify with the aid of . (2.17) If we evaluate the residue of J n at y = 0, it returns to which is also the completeness relation as all x i 's are trivially greater than zero. In fact, (2.13) and (2.17) can be trivially obtained, if we switch to the perspective which considers x 1 , . . . , x n < y and x 1 , . . . , x n > y respectively instead. Such an equivalent but much simpler approach can be further generalized to as well as where we have used the expressions in (2.8). A mixed product of these two types is, for example, .
From these formulas of co-positive products, it is easy to observe that: for n forms that impose positivity conditions on a number of variables and these conditions involve only one common variable, denoted by y for instance, we have I 1 ∩ · · · ∩ I n = I 1 × · · · × I n × y n−1 , which is trivial to prove from the perspective above. When there are two or more common variables, this simplification is no longer valid in general. Two such examples are given below: , (2.23) as well as .

(2.24)
It is known that a d log form can be interpreted as the projective volume, which establishes its relevance with forms. But in our context, the concept of volume appears more often as the cancelation of spurious poles, for amplitudes or integrands that are rational functions. This is in fact the key mechanism making summing different terms from the triangulation of amplituhedron possible.

The Trick of Intermediate Variables at 3-loop
Now, we are ready to introduce the trick of intermediate variables to handle the 3 intertwining positivity conditions of the 4-particle amplituhedron at 3-loop. This is not the final answer that we pursuit, but it divides a difficult problem into two parts in a pedagogical way, and it is a nice mathematical warmup for the more precise description. These positivity conditions are Without loss of generality, let's work in the ordered subspace X(123) so that x 1 < x 2 < x 3 . Then D 12 > 0 unambiguously demands for instance. Depending on the sign of (y 2 −y 1 )(w 1 −w 2 ), we have where c 12 and c 21 are defined as the positive intermediate variables.
The corresponding forms are then in ordered subspaces (Y (12)W (21)+Y (21)W (12)) and (Y (12)W (12)+Y (21)W (21)) respectively, and the symbols Z + and Z − are related by the completeness relation here the identity I 12 denotes no positivity condition is imposed on z 1 , z 2 .
Therefore, in subspace X(123), for D 12 , D 13 , D 23 > 0 we need to figure out the product where the products involving y-and w-space are easy, so we mainly focus on the products of Z ± 's. There are 2 3 = 8 such triple co-positive products, as listed below: and now we will determine them one by one.
As c 31 is treated as a positive variable, instead of a rational function of other positive variables as it actually should be, the discussion above leads to the sum note that we have divided the c 21 -space. This leads to .
(3.10) In general, we find that for c ij , c jk , c ik with respect to T 1 , T 2 , T 4 , T 5 , T 7 , T 8 , it is most convenient to divide the c ik -space. While for c ij , c jk , c ki with respect to T 3 , T 6 , there is no need to divide any of them.
Then for T 3 , since z 1 > z 2 +c 21 > z 3 +c 32 +c 21 already implies z 3 < z 1 +c 13 , Z − 13 becomes redundant, which leads to now we see indeed there is no need to consider any c ij .
Then for T 4 , it demands that z 3 is less than both (z 1 +c 13 ) and (z 2 +c 23 ) while z 1 and z 2 are restricted to the subspace of z 1 > z 2 +c 21 . Analogously, we have the following discussion: Then for T 5 , z 3 +c 32 < z 2 < z 1 +c 12 implies that (z 1 +c 12 ) must be greater than (z 3 +c 32 ) while z 1 and z 3 are restricted to the subspace of z 1 > z 3 +c 31 . Analogously, we have the following discussion: which leads to .
(3.15) Then for T 6 , similar to T 3 , there is no need to consider any c ij , the sum is simply .
Then for T 7 , similar to T 5 , (z 1 +c 12 ) must be greater than (z 3 +c 32 ) while z 1 and z 3 are restricted to the subspace of z 3 < z 1 +c 13 . Analogously, we have the following discussion: .
(3.20) Now we have known all eight T i 's. A consistency check via the completeness relation gives where we have used (2.22), similarly we also have (dropping all 1/c ij prefactors) These relations, in fact, serve as an equivalent approach to obtain all other T i 's one by one after we know T 1 and T 3 , following the sequence below: In addition, we have also observed that from all other T i 's can be obtained via flipping c ij to −c ji in the denominator with respect to flipping each Z − ij to Z + ji , as well as setting c ij to zero in the numerator. Therefore, T 8 is named as the master form.
There is still another equivalent approach to get the master form which divides the z-space instead of the c-space. Defining we find the sum is then as expected. Both ways to get the master form using the completeness relations and dividing the z-space can be generalized beyond 3-loop. Once it is known, we can apply the observation above to get all 2 L(L−1) 2 co-positive products of arbitrary Z ± 's. This observation has not been proved, but it turns out to be valid at 4-loop. In appendix A, we use the latter way to get the master form at 4-loop and after that, we check this observation explicitly via two examples, as a mathematical exercise of curiosity.

A Naive Sum
Next, we continue to sum the former results over all ordered subspaces, and we find this naive sum which takes the advantage of intermediate variables "almost" reaches the correct answer, as it can reproduce 96 out of the total 120 monomials in the latter.
To figure out the co-positive products involving y-and w-space, we define, for instance: in which the orderings of y 1 , y 2 and w 1 , w 2 are the same or opposite respectively. According to (3.6), each Z + ij is associated with an S(ij), as well as Z − ij with an A(ij). Then we explicitly figure out the products of S's and A's with respect to all T i 's as note that in particular, above we have used Y (13) ∩ Y (32) ∩ Y (21) = 0 and so forth. These results are for subspace X(123) only, and we need to consider all other ordered subspaces of x, such as where switching 2 ↔ 3 for x, y, z, w leads to switching T 3 ↔ T 5 and T 4 ↔ T 6 as can be easily verified, and the rest pieces are similarly given by where (Correct answer) × Denominator as well as and we have defined the product of all physical poles as Denominator ≡ x 1 x 2 x 3 y 1 y 2 y 3 z 1 z 2 z 3 w 1 w 2 w 3 D 12 D 13 D 23 . (4.8) Since D ij contains 8 monomials, the correct answer has (2×8+4)×6 = 120 monomials, and the sum has 96 so their difference has 4×6 = 24 monomials. It is important to notice that terms such as x 2 x 3 z 1 z 2 (−y 1 w 1 ) are not dual conformally invariant by themselves, but grouped as x 2 x 3 z 1 z 2 D 13 they are. This tentative answer simplified by the trick of intermediate variables captures more than we expect. Even though it oversimplifies the complexity of c ij 's which are functions of x, y, w, it still gives most parts of the correct answer. If we manually heal the dual conformal invariance, it is then correct. Remarkably, even if it does not give the full numerator, it can wipe off the subspace division of all positive variables, which frees it from spurious poles. After we refine the calculation in order to reach the correct answer, we will return to discuss the diagrammatic interpretation of (4.6).

Refined Co-positive Products
To precisely describe the 4-particle amplituhedron at 3-loop, we need to further refine co-positive products for each ordered subspace of y and w, based on the former discussions using intermediate variables. These seemingly lengthy results can be nicely rearranged in order to manifest its simple mathematical structure, namely the Mondrian diagrammatic interpretation.
From the previous setting we know that, for each ordered subspace of x, there are eight T i 's, namely the co-positive products in terms of intermediate variables. From (4.2), each of T 1 , T 8 corresponds to six ordered subspaces of y and w, while each of T 2 , T 3 , T 4 , T 5 , T 7 , T 8 corresponds to four, so that in total their number is 6×6 = 36 as expected. If we abandon intermediate variables, in principle we have to figure out 36 co-positive products instead of 8, as elaborated in the following.
For T 1 , the six different ordered subspaces lead to six different sets of c ij 's. First for Y (123)W (123), the condition c 31 > c 32 +c 21 is now replaced by (y 32 + y 21 )(w 32 + w 21 ) x 32 + x 21 > y 32 w 32 where x 32 , x 21 , y 32 , y 21 , w 32 , w 21 are positive variables in this subspace (as usual, we first work in X(123)).

The Correct Sum and its Mondrian Diagrammatic Interpretation
Collecting the 36 co-positive products for all ordered subspaces of y and w, we can continue to sum these results, and this time, it indeed reaches the correct answer. Instead of a brute-force summation, for each piece we delicately separate the contributing and the spurious parts. The former manifest the Mondrian diagrammatic interpretation with which they nicely sum to (4.6), while the latter sum to zero at the end. = (prefactors) × (x 21 x 32 z 1 z 2 D 13 + x 21 x 32 z 1 z 2 (y 21 w 32 + y 32 w 21 )), where again we will drop the prefactors that simply encode its information of ordered subspaces as well as physical poles. The first term above denotes the seed diagram which pictorially is a horizontal ladder, the first diagram given in figure 2. According to the contact rules conceived in the introduction, since boxes 1, 2 have a horizontal contact and so do boxes 2, 3, we can trivially read off the factor x 21 x 32 z 1 z 2 D 13 from that ladder diagram. In fact, this factor originates from x 21 x 32 z 12 z 23 D 13 in the ordered subspace Z(321), before we sum over all subspaces of z that admit it. As we have known Y (123)W (123) forbids any vertical contact of boxes (or loops), so we only have a horizontal ladder for this subspace, while the rest terms are spurious. For later convenience, we can define for W (321). It is clear that for different orderings of w, although their positive variables are different, the factors corresponding to any contact between boxes are the same. For example, both W (213) and W (231) admit the third diagram in figure 2, so the relevant w factors are w 12 and (w 13 +w 32 ) respectively, both of which equal to (w 1 −w 2 ). We also see that Y (123)W (321) admits all six diagrams, since the orderings of y and w are completely opposite. Let's sum the six spurious parts over subspaces of w for Y (123), which gives and as usual the prefactors are dropped. For the sum of each seed diagram over all subspaces that admit it, we will present examples of two distinct topologies below. First, for the first diagram in figure 2, x 21 x 32 z 1 z 2 D 13 trivially remains the same after we sum it over subspaces of y and w, since the completeness relation gives then it becomes x 2 x 3 z 1 z 2 D 13 after we sum it over subspaces of x that admit it, since admitting X x 21 x 32 = X(123) x 21 x 32 = 1 and this is the correct answer as one of those in (4.6). Then, for the third diagram in figure 2, x 31 x 32 z 1 z 2 y 21 w 12 becomes x 2 3 z 1 z 2 y 2 w 1 since admitting Y admitting W y 21 w 12 = 1 y 3 w 3 Y (12)W (21) y 21 w 12 = 1 y 1 y 2 y 3 w 1 w 2 w 3 y 2 w 1 , (6.11) as well as admitting X x 31 x 32 = X(σ(12) 3) x 31 x 32 = 1 and this is another one in (4.6). The rest four diagrams of different orientations are similar. We can continue the separation for the rest five orderings of y, each of which contains six orderings of w. Since we still work in X(123), the general seed diagrams for different orderings of y are given in figure  3, where some boxes are kept blank as the ordering of x alone can only fix part of numbers filled in these boxes. Straightforwardly, for Y (132) we have

Summary: a Mondrian Preamble
By separating the contributing and the spurious parts of each form in all ordered subspaces and assigning the former with corresponding Mondrian factors, which follow simple rules given by between any two loops labelled by i, j, we obtain the seed diagrams. If we assume the spurious terms will always sum to zero at the end, there is no need to sum the seed diagrams over all ordered subspaces since they are already topologically valuable. There is a simple way to find seed diagrams: let's work in simply one ordered subspace X(12)Z(21)Y (12)W (21) at 2-loop, as the first nontrivial example. Then, it is clear that D 12 is trivially positive so there is no positivity condition to be imposed. But as a physical pole D 12 must appear in the denominator, which identically turns the form into As usual, dropping the prefactors which contain all physical poles, we precisely obtain two 2-loop ladders of horizontal and vertical orientations (the vertical one is shown in figure 1). The 3-loop example is more interesting. Similarly in ordered subspace X(123)Z(321)Y (123)W (321), we can separate the triple product as D 12 D 13 D 23 = x 21 z 12 · x 32 z 23 · D 13 + y 21 w 12 · y 32 w 23 · D 13 + x 31 z 13 · x 32 z 23 · y 21 w 12 + x 21 z 12 · x 31 z 13 · y 32 w 23 + x 21 z 12 · y 31 w 13 · y 32 w 23 + y 21 w 12 · y 31 w 13 · x 32 z 23 , which precisely correspond to the six diagrams in figure 2 (including two ladders and four tennis courts). Here, for notational compactness we have defined x 31 ≡ x 32 +x 21 for instance, as x 32 and x 21 are primitive positive variables in this subspace while x 31 is not.
In general, Mondrian diagrams of higher loop orders satisfy this neat pattern: the product of all D ij 's can be expanded as a sum of all topologies and orientations in an ordered subspace of which the orderings of x, z are completely opposite, so are those of y, w. However, there are more subtle issues to be clarified, and we will continue to discuss them more systematically in the subsequent work.