All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity

This article introduces a systematic framework to understand (not to derive yet) the all-loop 4-particle amplituhedron in planar N=4 SYM, utilizing both positivity and the Mondrian diagrammatics. Its key idea is the simplest one so far: we can decouple one or more sets of loop variables (x,y,z,w) from the rest by just setting these variables to either zero or infinity so that their relevant positivity conditions are trivialized, then the all-loop consistency requires that we get lower loop amplituhedra as"residues". These decoupling relations connect higher loop DCI integrals with the lower ones, enabling us to identify their coefficients starting from the 3-loop case. And surprisingly, the delicate mechanism of this process is the simple Mondrian rule D=X+Y, which forces those visually non-Mondrian DCI integrals to have the correct coefficients such that the amplituhedron can exactly reduce to the lower loop one. Examples cover all DCI integrals at L=3,4,5,6, especially, the subtle 6-loop coefficients +2 and 0 are neatly explained in this way.

as we will introduce a new systematic framework to more clearly integrate positivity with the Mondrian diagrammatics.
To begin to rebuild everything, forgetting all later advances, we can return to the original definition of this problem [1,2]. First let's introduce a convenient convention for the following derivations: we will use the dimensionless ratio as the integrand, for example, in the 2-loop integral [2] dx 1 x 1 R is the integrand we will extensively manipulate. In other words, the full integral is made up of the d log measure of all positive variables and this ratio R. In particular, the 1-loop integrand is trivially 1 as there is no mutual positivity to be imposed. With this convention, when the integral is evaluated at either zero or infinity with respect to some variables, there is no extra factor to be added to the residual integrand, and the d log measure of those unfixed variables can be dropped for convenience.
Then what use does this residue evaluation at zero or infinity have, which seems trivial compared to the positive cuts [5]? If we simply set x j → ∞ and z j = 0 in

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D ij becomes trivially positive, since the positivity of z i is magnified by a positive infinity factor. Let's be more concrete and immediately look at the 3-loop case: x 3 → ∞ and z 3 = 0 lead to so D 13 , D 23 are positive, and we may claim that the third loop "decouples" from the rest two loops while positivity of D 12 remains to be imposed. Now according to the integrands defined above (in terms of the 3-loop result given in [3]), namely R 2 = x 2 z 1 + x 1 z 2 + y 2 w 1 + y 1 w 2 D 12 , (1.5) R 3 = x 2 x 3 z 1 z 2 + y 2 y 3 w 1 w 2 D 12 D 23 + x 2 3 z 1 z 2 y 2 w 1 + x 2 x 3 z 2 1 y 3 w 2 + x 2 z 1 y 2 3 w 1 w 2 + x 3 z 2 y 2 y 3 w 2 1 D 12 D 13 D 23 for L = 2, 3, the residue of R 3 at x 3 = ∞, z 3 = 0 is exactly R 2 ! This simple relation reflects the consistency of 4-particle amplituhedron as expected. We can further make it a bit more nontrivial by similarly setting where the infinitesimal ε is used to characterize the divergence of both x 3 and y 3 . Now the same relation R 3 → R 2 also holds but in a more interesting way as we will explain. Both situations above in fact encode the Mondrian diagrammatics [4]: in the first case a rectangle-like loop is removed, while in the second a corner-like loop is removed, as visualized in figure 1. For the rectangle removal, as shown in the 1st line of figure 1, two examples of non-vanishing contributions are given, from which the reduction from R 3 to R 2 can be transparently seen. The non-vanishing criterion of a diagram is, its third loop must have contacts with the external faces y, z, w (here we keep using the convention in [3,4,5], namely faces x, z, y, w locate at the left, right, top and bottom of the diagram respectively). Similarly for the corner removal, a diagram must let its third loop have contacts with the external faces z, w in order to be non-vanishing. Of course, the rectangle or corner removal can have different choices of orientation by dihedral symmetry but without loss of generality, we stick to y, z, w or z, w for consistency as above. However, unlike the rectangle removal for which each 3-loop diagram simply reduces to a 2-loop one, the corner removal leads to more interesting relations among different 3-loop diagrams, as they together reduce to a 2-loop counterpart. As shown in the 2nd line of figure 1, after removing the third loop, these three diagrams reduce to the same 2-loop diagram but with various prefactors, of which the sum is unity: this is exactly the completeness relation in Mondrian diagrammatics [4]: Diagrammatically Mondrian factors X i , Y i , D i mean loop 3 has a horizontal contact, vertical contact or no contact with loop i = 1, 2 respectively. From these easy examples of two types of loop removal, we see how the Mondrian diagrammatics helps understand the interconnections among different diagrams of various DCI topologies (including their coefficients) in an extremely neat way.

Nontrivialities at 4-loop
Next, we are curious to see how this wishfully simple mechanism deals with the more sophisticated 4-loop case, since it involves DCI topologies with coefficient −1 and non-Mondrian pole structure, both of which are absent at lower loop orders. First of all, let's recall the 4-loop DCI topologies as given in figure 2.   [6]). The red curve denotes a rung rule factor D ij , while rung rule factors x, y, z, w are diagrammatically suppressed for convenience.
Here topologies T 1 , . . . , T 7 are Mondrian while T 8 is not. As we have known, T 7 and T 8 are associated with coefficient −1, and T 1 , . . . , T 6 are associated with +1, moreover, T 4 has a D ij factor in the numerator of its integrand. To see why these coefficients are so, we can pretend that they are still unknown yet and denote them by s 1 , . . . , s 8 . Immediately, we can perform the rectangle removal of loop 4. More rigorously speaking, we impose the limit in the 4-loop integrand R 4 , which takes into account DCI loop integrals of all possible orientations given by dihedral symmetry (the number of which can be 8, 4, 2, or 1 for each topology) and all 4! permutations of loop numbers. Then in the expansion the leading term R 4 (0) depends on s 1 , s 2 , s 3 , s 4 , s 5 only, and R 4 (0) = R 3 when s 1 = s 2 = s 3 = s 4 = s 5 = 1 as expected. From the Mondrian diagrammatic perspective, this is trivial to understand as a 4-loop example of the rectangle removal. In fact we can further generalize the rectangle to remove more loops at a time, which will justify the existence of T 6 , T 7 . As visualized in figure 3, we now remove a block containing loop 3, 4 by imposing note that ε helps regularize the expression (∞−∞) in D 34 and renders this factor vanish. Then loop 3, 4 decouple from the rest two loops, and in the expansion where the additional subscript 2 of R 4 denotes removing two loops at a time, we find which equals R 2 when s 3 = s 6 = 1 and s 7 = −1. This is also easy to understand diagrammatically, and the interesting combination (2s 6 +s 7 ) explains why we need a minus sign for the cross topology T 7 : the block removal of loop 3, 4 of two different orientations of T 6 gives the same 2-loop diagram, therefore one must be eliminated in order to maintain R 4,2 (0) = R 2 while all orientations of T 6 and T 7 are used exactly once, as shown in figure 3. This cancelation mechanism is somehow analogous to the cancelation between the cross and the brick-wall patterns in Mondrian diagrammatics [4], and we will see more examples at higher loop orders. Now only s 8 awaits to be explained and we must use the corner removal to detect this non-Mondrian topology T 8 , since it has no rectangle or block to be properly removed. Similarly, for removing loop 4 we impose the limit then in the expansion R 4 (ε) = R 4 (0)+O(ε), we find to maintain the consistency we must take s 8 = −1. This is easy to understand if we look at T 4 , T 8 together among others, as the D ij factor in the numerator of T 4 's integrand requires a counter term for producing the correct Mondrian factor. More concretely, in the 1st line of figure 4 the relevant two diagrams give after using is the desired Mondrian factor, which characterizes the contacting relation between loop 4 in the diagram of topology T 4 and the 3-loop sub-diagram (or the resulting 3-loop diagram at the RHS). Moreover, the cancelation between T 6 and T 7 in the 4-loop corner removal R 4 → R 3 again holds, as the relevant three diagrams in the 2nd line of figure 4 lead to the combination which is exactly isomorphic to the cancelation between the cross and the brick-wall patterns in Mondrian diagrammatics [4]. Let's summarize the nontrivialities in the 4-loop case via understanding s 1 , . . . , s 8 . First, it is useful to generalize the rectangle removal to the block removal, in order to check the consistency of decoupling more than one loop at a time. At 6-loop order we will also need the block removal of three loops, and so on. Next, topologies T 7 , T 8 with coefficient −1 serve as counter terms of those with +1, and while T 7 has a clear meaning in Mondrian diagrammatics, T 8 appears to be the necessary company of T 4 which has a nontrivial D ij factor in its integrand. At 5-loop order and higher, even a company topology or a group of company topologies will have its further company, enriching the diversity of contributing topologies while maintaining the overall Mondrian consistency.

Rectangle and Block Removals at 5-loop
To see more nontrivial examples of various patterns found at L = 3, 4, and to check whether new features or exceptions appear, we move on to the 5-loop case. First let's recall the 5-loop DCI topologies as given in figure 5, where all 34 topologies are reorganized for later convenience while the same labels used in [5] are kept. Upon these inputs, we find the rectangle removal of loop 5 leads to R 5 (0) = R 4 as expected. And the block removal of loop 4, 5

Identifications of the Rest Coefficients
First of all, s 28 , s 29 , s 30 can be trivially determined by the 4-loop knowledge. Obviously, T 28 is the company topology of T 10 , similar to the fact that T 8 is the company topology of T 4 in the 4-loop case, as the T 10 , T 28 pair is the counterpart of the T 4 , T 8 pair plus one rung. Similarly, T 29 , T 30 are the company topologies of T 19 , note that T 29 has one rung rule factor and T 30 has two substitution rule factors which result from the corresponding rung rule factor of T 19 . Therefore we simply have s 28 = s 29 = s 30 = −1. Now let's consider s 15 , s 16 which are a bit tricky. To separate s 15 , s 16 from other unknown coefficients, we impose as many cuts as possible around the rim of a particular T 15 diagram, as shown in figure 6, and then use a special two-step expansion. To disentangle s 15 and s 16 , we also relax one external cut, namely y 1 here, and the reason to do so will be clear shortly. Explicitly, we impose the limit then in the two-step expansion (note the order of expansions matters as we intentionally utilize the x 5 , z 4 factors of this diagram to separate it from other sub-leading contributions) where the prime denotes additional cuts x 3 = w 3 = z 1 = 0 besides removing loop 4,5 as indicated in figure  6, we find To maintain the consistency we must have s 15 +s 16 = 0 and 2s 15 +s 16 = 1, so s 15 = 1 and s 16 = −1, which explains why y 1 must be non-vanishing, otherwise we cannot identify s 15 , s 16 with merely 2s 15 +s 16 = 1. As we have assumed 2s 15 +s 16 +2s 32 +s 33 = 0 in the previous section, this condition reduces to 2s 32 +s 33 = −1 which awaits to be confirmed. Next, we can identify s 31 , s 32 , s 33 upon the inputs of s 15 , s 16 , s 30 in a similar way. Picking a particular T 32 diagram as given in figure 7, we impose the limit and note the external cut of y 1 is relaxed, then in the expansion R 5 (ε) = R 5 (0)+O(ε) we find so the consistency requires s 31 = s 32 = −1 and s 33 = 1. Again, y 1 must be non-vanishing, otherwise we can merely know one condition. Now we have confirmed 2s 32 +s 33 = −1 and hence 2s 15 +s 16 +2s 32 +s 33 = 0. Then for s 27 , we can pick a particular T 27 diagram as given in figure 8 and impose the limit now upon the inputs of s 15 , s 16 and s 28 , . . . , s 33 , we find therefore s 27 = −1. Finally for s 34 , we can pick a particular T 34 diagram as given in figure 9 and impose the limit upon the inputs of s 15 , s 16 and s 30 , . . . , s 33 we find

Nontrivial Mondrian Diagrammatic Relations at 5-loop
Knowing all these coefficients, we then proceed further to understand them, for example, why a coefficient is −1 instead of 1, which should not be just an incidental result of imposing cuts at either zero or infinity. Once we find the Mondrian interconnections among various DCI topologies, their coefficients will become a natural consequence of the simple relations extracted from previous derivations. More concretely, we would like to explain the universal decoupling relation using one corner removal, which, unlike the rectangle or block removal, can cover all topologies. The limit to be imposed is simple: namely removing loop 5, then in the expansion R 5 (ε) = R 5 (0)+O(ε) we find R 5 (0) = R 4 . However, this is not the end of the story since R 5 (0) = R 4 is a redundant relation and it can be further dissected into many much more transparent sub-relations, as diagrammatically shown in figures 10 and 11. In figure 10, all nontrivial corner removals of 5-loop DCI topologies are indicated. We will focus on groups of corners denoted by 1, . . . , 7, while groups denoted by A, . . . , F are the direct extensions of the corner removals of T 4 , T 8 in the 4-loop case (see figure 4). The rest unmentioned corners are as trivial as the 3-loop corners or 4-loop corners except those of T 4 , T 8 , as they locate in visually Mondrian topologies and do not involve the D ij factor.
In figure 11, groups 1, . . . , 7 are further separated into three types so that we can more clearly understand their nontrivialities, let's select one example from each type to elaborate these delicate relations. In the 1st diagrammatic equality, the 1st and 3rd diagrams have Mondrian pole structures, and in these two diagrams the removed loop has horizontal contacts with loop 1,2 while its contact with the unlabeled loop on top of loop 1 is horizontal in the 1st diagram and vertical in the 3rd. Naively this should give according to the definitions in (1.9), which uses a Mondrian completeness relation and here 3 denotes the unlabeled loop. However, the D ij factors in the 1st and 3rd diagrams complicate this relation and that's why we also need the 2nd and 4th diagrams with minus signs to offset that, then we can exactly get the neat result at the RHS with Mondrian factor X 1 X 2 . The 2nd and 3rd diagrammatic equalities share the same feature of needing non-Mondrian company topologies, to offset the extra complexity brought by the D ij factors. However, such a company topology does not have one-to-one correspondence to a particular Mondrian topology, unlike the T 4 , T 8 pair at 4-loop.
For the 4th diagrammatic equality, under the corner removal, schematically it is proportional to X + Y −D, so it simply vanishes. The 5th equality follows exactly the same cancelation mechanism, though it is not so obvious as the 4th.
For the 6th diagrammatic equality, under the corner removal three non-Mondrian diagrams sum to a 4-loop non-Mondrian diagram, due to Though the resulting 4-loop diagram is not Mondrian, its contact with the 5th loop is still Mondrian, so that we can use the Mondrian completeness relation. The 7th equality is similar but more nontrivial as note the 2nd diagram has a plus sign so it contributes a minus in the Mondrian completeness relation, as the resulting 4-loop diagram also has a minus sign. The Mondrian factor D 1 X 2 Y 3 from this diagram is not obvious in the sense of horizontal and vertical contacts, but we can deform its external profile to manifest this, as shown in figure 12. Now the external profile of this 5-loop diagram is not a rectangle, but we can see a familiar 4-loop non-Mondrian diagram hidden in it. With Mondrian factor D 1 X 2 Y 3 clarified, which is the desired result for offsetting two D 1 X 2 Y 3 factors from the rest four diagrams, we see the Mondrian diagrammatics works more effectively than naive visual intuition. A final remark is, the 2nd diagram also serves as a company topology of the rest four, similar to the complexity of the first three identities.

Coefficients +2 and 0 at 6-loop
Finally we can take a glance at the 6-loop case, by investigating the coefficients of two special 6-loop DCI topologies. First of all, the 6-loop amplituhedron or integrand involves 229 non-vanishing contributions of DCI topologies, among which 125 have Mondrian pole structures as listed in Appendix B of [4], while the rest 104 non-Mondrian ones are the company topologies in the sense of corner removal. Interestingly, we also need six vanishing DCI topologies, namely those with coefficient 0, for a complete understanding of the 6-loop corner removal. All these topologies with coefficients can be found in the original result [8].
Similar to the 5-loop case we have extensively described, the new Mondrian diagrammatic relations at 6-loop also can be separated into three types: those with obvious Mondrian pole structures, the vanishing or canceling ones and those of non-Mondrian company topologies. Now we consider two particular 6-loop DCI topologies with three relevant Mondrian diagrammatic relations, as shown in figure 13. The first topology is the 2nd diagram in the 1st relation, or the 2nd diagram in the 2nd relation, as each of them serves as a company topology of the 1st diagram in the 1st or 2nd relation, following exactly the same mechanism of T 4 , T 8 pair at 4-loop (see figure 4). Since this topology appears twice and in both situations it has a plus sign, its overall coefficient is simply +2 as we add up these two pluses! Similarly, the second topology is the 3rd diagram in the 1st relation, as a company topology for the other corner of the 1st diagram, or the 5th diagram in the 3rd relation, and note that they belong to the same topology though drawn differently. Now in both situations it has a plus and a minus, so these two signs cancel and its overall coefficient is 0! One may find the 3rd relation unfamiliar, but it is simply the 7th relation in figure 11 if we remove the 6th loop as indicated in figure 13. Imagine the deformation in figure 12 to better visualize this analogy, one will find this 6-loop relation completely trivial based on its 5-loop counterpart with an overall sign reverse for all topologies. Then the two special coefficients +2 and 0 are neatly explained.
At 7-loop order there is no novelty other than +2 and 0 coefficients [8], while starting from the 8-loop case fractional coefficients begin to appear [9,10]. Therefore we expect a nontrivial generalization of the Mondrian diagrammatic relations at L ≥ 8, but they should be not too exotic since these coefficients are still rational. Finally, we would like to explore how the Mondrian consistency connecting amplituhedra of different loop orders can be extended to the generic case of more than four particles [11,12,13], or what it can tell us about the latter from the 4-loop knowledge [14].