Triangulation-free Trivialization of 2-loop MHV Amplituhedron

This article introduces a new approach to implement positivity for the 2-loop n-particle MHV amplituhedron, circumventing the conventional triangulation with respect to positive variables of each cell carved out by the sign flips. This approach is universal for all linear positive conditions and hence free of case-by-case triangulation, as an application of the trick of positive infinity first introduced in 1910.14612 for the multi-loop 4-particle amplituhedron. Moreover, the proof of 2-loop n-particle MHV amplituhedron in 1812.01822 is revised, and we explain the nontriviality and difficulty of using conventional triangulation while the results have a simple universal pattern. A further example is presented to tentatively explore its generalization towards handling multiple positive conditions at 3-loop and higher.


Introduction
The amplituhedron proposal of planar N = 4 SYM [1,2] is a novel reformulation which only uses positivity conditions for all physical poles to construct the amplitude or integrand. For given (n, k, L) where n is the number of external particles, (k+2) is the number of negative helicities and L is the loop order, the most generic loop amplituhedron is defined via here C αa is the (k×n) positive Grassmannian encoding the tree-level information and D (i)αa is the (2×n) positive Grassmannian with respect to the i-th loop, and Z I a is the kinematical data made of n generalized (k+4)-dimensional momentum twistors, which also obeys positivity as Z a 1 . . . Z a k+4 > 0 for a 1 < . . . < a k+4 . (1.2) To pedagogically investigate this object, we often separately consider the "pure loop part" of which k = 0, namely the MHV sector [3,4], in particular the 4-particle case (n = 4) has been extensively understood up to high loop levels [5,6] and can be compared to the known results from 2-loop to 10-loop level [7,8], and the "pure tree part" of which L = 0 [9,10,11], as well as the simplest nontrivial mixture of these two: the 1-loop NMHV case (k = 1, L = 1) treated in [12]. Below we will focus on the 2-loop MHV amplituhedron determined by the sign flips [13] plus a single mutual positive condition, and the relevant background of its integrand can be found in [14,15,16]. There are also interesting investigations of loop amplituhedron and the integrated results [17,18].
Back to the specified object of interest, now we continue to elaborate following the general definition above. As k = 0, there is no Y = C ·Z part, and L = 2 gives two L = D·Z parts, each of which individually obeys physical constraint (note the twisted cyclicity Z n+1 = −Z 1 for k = 0) and together they obey the mutual positive condition First, to triangulate the trivial 1-loop MHV amplituhedron and identify each cell, we need to impose the sign-flip constraint [13], namely in the sequence (defining L (1) ≡ AB and L (2) ≡ CD below) AB12 + AB13 ± AB14 ± . . . AB1, n−1 ± AB1n +, (1.5) while the head AB12 and tail AB1n are both positive, the entire sequence has two sign flips, so there are (n−2)(n−3)/2 possibilities. Explicitly, if the two sign flips occur at which satisfies physical constraint ABZ a Z a+1 > 0 and the sign-filp constraint, similar for L (2) = CD: where x 1 , w 1 , y 1 , z 1 and x 2 , w 2 , y 2 , z 2 are all positive variables. Then, the nontrivial mutual positive condition of major concern is L (1) L (2) = ABCD > 0 (1.9) for each composite 2-loop cell made of any two 1-loop cells, so there are (n−2) 2 (n−3) 2 /4 combinations. Note that, there are two types of triangulation. The first type is the sign-flip triangulation to carve out each 1-loop cell with a specific parameterization, while the second is the triangulation with respect to positive variables of each cell, identical to those extensively manipulated in the 4-particle case which has only one cell. In this work the triangulation mentioned is the second type, and we will see how the idea of positive infinity [6] can free us from this tedious task first at 2-loop, in an extremely simple way.

Minimal Review of Positive d log Forms and Dimensionless Ratios
To get familiar with the mathematical concepts we will extensively use, let's first give a minimal review of d log forms in positive geometry. As defined in [2], for a positive variable x without further restriction, we know its d log form (with integration over x) is which has a singularity at x = 0. If we require x > a, as the singularity is shifted to x = a, then the form is on the other hand, the form of x < a is defined as the complement of x > a (dropping the integration): which naturally has two singularities at x = 0 and x = a. This equality is also known as the completeness relation [5], if we reshuffle it as 1 furthermore if we drop the integration over d log x = dx/x instead of dx, we get the completeness relation in terms of dimensionless ratios [6], which is a more natural way to characterize positive d log forms. Here x and a are treated on the same footing (a also can be a variable), and the sum is always unity. As done in [5], we can generalize these conditions to n x i > a and n x i < a, and the corresponding dimensionless ratios also sum to unity: To inductively prove the dimensionless ratio of n x i > a is (2.7) we can first assume it holds for n−1 x i > a. Then depending on n−1 x i ≷ a, we require simply x n > 0 or x n > a− n−1 x i to satisfy n x i > a, which gives as expected. And a also can be generalized to a sum of positive variables, then the dimensionless ratio of which treats n x i and m y j on the same footing. If any x i goes to positive infinity [6], the ratio above trivially becomes 1, as any x i → ∞ trivializes n x i − m y j > 0. Now we are ready to move forward, to explore the extraordinary simplicity hidden in the 2-loop MHV amplituhedron.

Triangulation-free Trivialization for Linear Polynomials
For a single positive condition defining (a cell of) the 2-loop MHV amplituhedron, it's an ubiquitous fact that the numerator part of its relevant d log form is always "maximally positive", instead of just positive. For example, in the 4-particle case, if we look at the dimensionless ratio of its d log form which is the nontrivial factor in the full loop integral obviously D + 12 is the maximally positive part of D 12 , namely the term-wise positive polynomial including every positive term in D 12 . This pattern also applies to all n ≥ 5 particle cases, and usually the proof must be done case by case with triangulation. We are often annoyed by the fact that, the tedious triangulation is inevitable but still this process leaves no trace in the final sum, which however means the sum is correct. This subtle phenomenon motivates us to circumvent the triangulation, and maybe it is possible to redefine positive conditions that characterize the generic multi-loop MHV amplituhedron in this way.
First for a single positive condition, so far all cases we have encountered are linear in all variables. So we can assume this polynomial takes a not-so-general form as where {x i }, {y j }, {z k } are three subsets of all positive variables, and P 0 and P i are independent of any x i . Such an expansion is always possible, and we can further expand P i as Obviously, this nested expansion can be done for as many levels as needed, while this not-so-general form has only three levels of expansion but it is enough for an inductive proof. Let's give some examples: all of these are linear polynomials of level 1, 3, 3 respectively. Now assuming the positive sub-condition we want to determine its dimensionless ratio P i P i . (3.7) First, P i must also be linear in {y j }, because a y 2 j term will render this ratio diverge at y j = ∞, so will a 1/y j term at y j = 0. Recall that when the positive condition is trivialized, this ratio must be 1. Therefore we can take the following ansatz then if all y j = 0, we have a simple ratio Note that P i,0 is of level one without further nontrivial entanglement as assumed, for example which trivially leads to P i,0 = P + i,0 , as we have proved via (2.9) in the previous section. Next, inspired by the trick of positive infinity in [6], each y j = ∞ also leads to P i,j = P + i,j . Since P + i,0 , P + i,j are independent of any y j , we must have this "prime" operation is actually the familiar "positive terms extraction". Because the derivation above is inductive, similarly for we also have which finishes the clean proof of the dimensionless ratio for a linear P of any levels of nested expansion.

2-loop MHV Amplituhedron Revisited
Now for a generic cell of the 2-loop MHV amplituhedron [4,13], from the parameterization we see the mutual positive condition is a linear polynomial, and it can have maximally four levels. Let's see a concrete example by choosing i = 2, j = 8, k = 4, l = 6, (4.3) then this quantity becomes = C + x 1 (− C 2 + x 2 (− C 2,4 + y 2 C 2,4,6 + z 2 C 2,4,7 ) + w 2 (− C 2,5 + y 2 C 2,5,6 + z 2 C 2,5,7 )) + w 1 (− C 3 + x 2 (− C 3,4 + y 2 C 3,4,6 + z 2 C 3,4,7 ) + w 2 (− C 3,5 + y 2 C 3,5,6 + z 2 C 3,5,7 )) , where the positive determinants are defined as C = Z 1 , x 2 Z 4 +w 2 Z 5 , y 2 Z 6 +z 2 Z 7 , y 1 Z 8 +z 1 Z 9 , we see that it actually has three levels, since C i,j,k is trivially positive and needs not expand as a fourth. Then we can immediately apply the proof in the previous section to show that its dimensionless ratio is note that if we try to prove this result with triangulation, it will be extremely tedious already for the 2-loop case, as we have to handle complicated shifting and intersecting relations in three copies of 2-dimensional planes spanned by variables (x 1 , w 1 ), (x 2 , w 2 ) and (y 2 , z 2 ). Since such a proof holds for generic i, j, k, l, all d log forms corresponding to various cells of the 2-loop MHV amplituhedron are trivialized and free of the case-by-case triangulation.

Proof in 1812.01822 Revised
However, the proof in [4] using triangulation also seems clean (see its Appendix B). Now we explain why this proof should be revised while its conclusion still holds. There, the positive condition of combination i < k < l < j is reorganized as (the arguments indicate how a, b, c, d, e depend on z 2 , w 1 , x 1 , w 2 , y 2 ) then the subsequent discussion continues as if these a, b, c, d, e were all constants. We find it problematic, because this is equivalent to rescaling z 2 , w 1 , x 1 , w 2 , y 2 by five constants respectively, but the Jacobian of this rescaling is not trivially 1. So why is the conclusion still correct?
The subtle secret here is that though the rescaling is illegal, a, b, c, d, e are still positive. So pretending that they were constants just gives us the same result while the correct logic is not so trivial. Without the trick of positive infinity, we will have a tough work of triangulation to do. Now we find an even cleaner and also more general proof for this neat result.
as expected. Furthermore, in [5] there are other seven d log forms (namely T 1 . . . T 7 ) that can be obtained by flipping c ij to −c ji in the denominator and setting c ij to zero in the numerator, which exactly reflects the logic of the new proof as the numerator always collects positive terms only. This example also provides a tentative approach to extend the triangulation-free trivialization to the cases with multiple positive conditions, as will be discussed more in the next section. But of course, we should note this example is a much simpler case in the context of 4-particle amplituhedron, as restricted to the ordered subspace X(123) in which x 1 < x 2 < x 3 . In general, n ≥ 5 particle cases at 3-loop will have various combinations of 1-loop cells in terms of three sets of loop variables, so the positive conditions are no longer uniform, and they may have more complicated nested expansions.

Outlook
The discussion above is also a key motivation to develop a triangulation-free approach, otherwise even the 3-loop work will be overwhelmingly difficult. The luxurious ambition is to extend the 2-loop proof to the all-loop, generic n-particle MHV amplituhedron, or directly redefine this geometric object with positivity but without the annoying triangulation. Here, we can easily trivialize positivity by evaluating the integral at zero or positive infinity with respect to some variables, however, unlike the 2-loop case, its challenge is to reconstruct the correct integrand or dimensionless ratio from multiple positive conditions. How to find a minimal set of such "cuts" that can fully cover every facets of the object, requires a further geometric understanding, especially about the shifting and intersecting relations among multiple higher dimensional planes representing the positive constraints.
Naturally, the 4-particle amplituhedron at 3-loop is a simplest nontrivial testing ground for this goal of which the result has been well known from various perspectives, and more importantly, in the 4-particle case, the positive conditions are always uniform and this symmetry is partly maintained upon the cuts. In fact, the Mondrian reduction [6] is a special type of application of these cuts, but we must know the DCI integral basis first in that diagrammatic context, and now we would like to derive the basis as well from a more algebraic perspective, as for the generic n ≥ 5 particle case there is no simple insight similar to the Mondrian diagrammatics. In the future, we will focus on the 4-particle case up to higher loops as usual, as well as the tentative derivation of the 5-particle case at 3-loop, using the triangulation-free approach.