All-loop Mondrian diagrammatics and 4-particle amplituhedron

Abstract

Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics. By further simplifying and rephrasing the key relation of positivity in the amplituhedron setting, remarkably, we find a completeness relation unifying all diagrams of the Mondrian types for the 4-particle integrand of planar N = 4 SYM to all loop orders, each of which can be mapped to a simple product following a few plain rules designed for this relation. The explicit examples we investigate span from 3-loop to 7-loop order, and based on them, we classify the basic patterns of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral patterns. Interestingly, for some special combinations of ordered subspaces (a concept defined in the previous work), we find failed exceptions of the completeness relation which are called "anomalies", nevertheless, they substantially give hints on the all-loop recursive proof of this relation. These investigations are closely related to the combinatoric knowledge of separable permutations and Schröder numbers, and go even further from a diagrammatic perspective. For physical relevance, we need to further consider dual conformal invariance for two basic diagrammatic patterns to correct the numerator for a local integrand involving one or both of such patterns, while the denominator encoding its pole structure and also the sign factor, are already fixed by rules of the completeness relation. With this extra treatment to ensure the integrals are dual conformally invariant, each Mondrian diagram can be exactly translated to its corresponding physical loop integrand after being summed over all ordered subspaces that admit it.

All-loop cuts from the Amplituhedron

“…A manifestly Yangian invariant diagrammatic formulation using so-called "momentum twistor diagrams" was introduced in [16] and used to study the structure of the one-loop geometry in [17]. The higher loop-level geometry of the amplituhedron was explored in detail in [18,19] and an attempt to completely understand the geometry at four points and progressively higher loops can be found in [20][21][22]. However, important open questions regarding the technical details of triangulating the amplituhedron remain.…”

4-particle amplituhedronics for 3-5 loops

“…Though the dominating principle is simple and symmetric up to all loops, as the loop order increases, its calculational complexity grows explosively due to the highly nontrivial intertwining of all L(L−1)/2 positivity conditions. So far the 4-particle amplituhedron has been fully understood up to 3-loop [3], from which we have incidentally found an intriguing pattern valid at all loop orders for a special subset of dual conformally invariant (DCI) loop integrals: the Mondrian diagrammatics [4]. Even though there still remain unknown characteristics of the connection between this neat formalism and down-to-earth physics, to say the very least, it offers us a much more efficient way for reorganizing the 3-loop results via a direct calculation, by extensively using the properties of ordered subspaces which further refine the space spanned by x, y, z, w.…”

“…Let's first briefly summarize some notions with relevant notations introduced in [3,4] which are frequently used in this work.…”

“…note that we have omitted the measure factor, following the convention of [3,4]. Originally, the full d log form is defined as…”

“…Each subspace admits some Mondrian seed diagrams [3], for example, X(123)Z(321)Y (123)W (123) admits the ladder diagram in figure 5, which can be characterized by a Mondrian factor X 12 X 23 D 13 , with X ij = (x j −x i )(z i −z j ), Y ij = (y j −y i )(w i −w j ) and D ij = X ij +Y ij . This factor is determined by the contact rules between any two loops defined in [3,4] as horizontal contact: X ij vertical contact: Y ij no contact: D ij (always taking i < j for D ij ) (1.6) For a particular subspace we can derive its d log form by demanding D 12 , D 13 , D 23 > 0. Then multiplying its form by all positive denominators gives its proper numerator, and the dimensionless ratio between this numerator and D 12 D 13 D 23 encodes the positivity constraints, which becomes 1 if the positivity is trivial.…”