Amplituhedron-Prime in Planar N=4 SYM

• Amplituhedron-Prime is a positive geometry defined by a union of sign-flip regions that decompose one-loop MHV amplitude integrands in planar maximally supersymmetric Yang-Mills theory.
• Its unique canonical form, shared with the original Amplituhedron, is expressed through dlog forms and triangulated via chiral pentagon and octagon structures.
• The framework offers the first evidence for a dual Amplituhedron, linking geometric decompositions directly to the dual structures in scattering amplitudes.

Amplituhedron-Prime is a positive geometry intimately connected to scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. It is closely related to, but geometrically distinct from, the standard Amplituhedron. Both objects share identical boundary stratifications and canonical forms but have disjoint interiors. The Amplituhedron-Prime plays a crucial role in providing an explicit decomposition of amplitude integrands into chiral geometric building blocks and establishes the first firm evidence for the existence of a “dual Amplituhedron.” Its rigorous definition and properties have been investigated across tree- and loop-level settings, yielding constructive triangulations and geometric insights not present in the original Amplituhedron framework.

1. Definition via Sign-Flip Sectors and Positive Geometries

The Amplituhedron-Prime, denoted $\mathcal{A}'$, is defined as a union of sign-flip-two and sign-flip-four regions in the loop Grassmannian, contrasting with the original Amplituhedron $\mathcal{A}$, which is a single sign-flip-zero region. For the one-loop MHV case ($k=0$, $L=1$) with $n$ points, external data consists of $n$ momentum twistors $Z_1, \dots, Z_n \in \mathbb{P}^3$ with all four-brackets positive. Let the loop variable be $(AB)$. The Amplituhedron $\mathcal{A}^{(n,0,1)}$ is the region

$$S^{(0)}_{\text{MHV}} :\quad \langle AB\,i\,i+1 \rangle > 0~\text{for}~i=1,\dots,n;\quad \langle AB\,\overline{ij} \rangle > 0~\text{for}~1 \leq i < j \leq n,$$

where $\mathcal{A}$0.

The Amplituhedron-Prime is the following union:

$\mathcal{A}$1

where $\mathcal{A}$2 denotes sign-flip-two regions of positive chirality and $\mathcal{A}$3 denotes sign-flip-four regions of negative chirality. Each region is defined by a prescribed pattern of sign changes in the sequence $\mathcal{A}$4(Herrmann et al., 2020).

Both $\mathcal{A}$5 and $\mathcal{A}$6 possess the same codimension-$\mathcal{A}$7 boundary stratification in $\mathcal{A}$8, though their geometric interiors are distinct. In higher generality, amplituhedron-like regions with non-maximal winding (i.e., $\mathcal{A}$9 in $k=0$0) fit naturally into this framework, and the union of all such regions forms the "squared amplituhedron"—a configuration relevant for canonical form squaring and dual geometry constructions(Dian et al., 2021).

2. Canonical Forms and Star Products

Despite their differing interiors, $k=0$1 and $k=0$2 share the same unique canonical top-form, which is the one-loop MHV integrand:

$k=0$3

with

$k=0$4

This same form can be expressed as a sum over canonical $k=0$5 forms associated with each sign-flip sector:

$k=0$6

where $k=0$7 is a box or degenerate chiral pentagon (sign-flip-two positive), $k=0$8 is the "chiral octagon" form (sign-flip-four negative), and $k=0$9 are parity-odd pentagon boxes.

At higher $L=1$0 and loop order, canonical forms for amplituhedron-like regions $L=1$1 are given by the star product of parity conjugate amplitudes (proven for $L=1$2 at tree level and for MHV at loop level):

$L=1$3

The star product $L=1$4 in bosonized superspace is defined by(Dian et al., 2021):

$L=1$5

where the precise structure depends on the bracketings of the involved invariants.

3. Triangulation by On-Shell Diagrams and Chiral Pentagons

A key structural feature of Amplituhedron-Prime is the existence of a triangulation into local geometric cells, explicitly realized by chiral pentagon and box forms at one loop, and by pairs of on-shell diagrams in general. In the canonical one-loop MHV expansion,

$L=1$6

Each $L=1$7 is the unique canonical form on a corresponding local positive region $L=1$8 or $L=1$9. These pentagon cells jointly cover $n$0, whereas they do not triangulate the original Amplituhedron $n$1; their union in $n$2 would leave spurious (nonphysical) boundaries.

More generally, amplituhedron-like geometries $n$3 can be triangulated by pairs of on-shell diagrams, allowing the canonical form to be constructed from products of Yangian invariants, validating the form $n$4(Dian et al., 2021).

4. Dual Polygon Picture and Internal Triangulation

On codimension-two boundaries—where the loop variable $n$5 is localized to a $n$6—both $n$7 and $n$8 project to polygons in this plane. There exists a duality map exchanging lines and points, under which the leading singularities (polygon vertices) of the original map to edges of the dual, and vice versa.

Remarkably, in this dual frame, chiral pentagon regions $n$9 are mapped to interior triangles which collectively triangulate the dual polygon. This structure is verified for $n$0 and conjectured to hold more generally. The dual of $n$1 is thus realized as an explicit internal triangulation of the putative dual Amplituhedron, formed by chiral building blocks. This provides the first explicit evidence for the existence of a dual Amplituhedron geometry(Herrmann et al., 2020).

5. Oriented Canonical Forms, Squared Amplituhedron, and Global Extension

Beyond individual winding sectors, one may form the "squared amplituhedron" $n$2—the union over all $n$3 with only proper-boundary inequalities and no sign-flip constraints:

$n$4

This union is not itself a positive geometry but a collection of almost disconnected $n$5 sectors, touching only at lower-codimension boundaries. The standard prescription for the canonical form fails due to multi-valued residues on sector boundaries; instead, a globally oriented canonical form is defined on the oriented Grassmannian double cover:

$n$6

For $n$7, this sum yields the explicit square of the amplitude's canonical form:

$n$8

Algorithmically, such forms can be computed by applying Cylindrical Algebraic Decomposition (CAD), patching $n$9 forms with sign rules dictated by orientation reversals(Dian et al., 2021).

6. Implications and Prospects for Dual Geometry

The existence and explicit triangulation of the Amplituhedron-Prime have several key implications:

• The chiral pentagon and octagon forms (for higher flips) are individually positive on the original Amplituhedron and serve as canonical building blocks for triangulating dual geometries.
• The decomposition of $Z_1, \dots, Z_n \in \mathbb{P}^3$0 into simple sign-flip regions provides a template for constructing dual positive geometries at higher $Z_1, \dots, Z_n \in \mathbb{P}^3$1 and higher-loop order, with a suggested progression to octagons, decagons, etc.
• The internal triangulation of the dual polygon via images of chiral pentagons supports the hypothesis that a dual Amplituhedron exists and that its canonical form is assembled from these local objects.
• A zero-form space $Z_1, \dots, Z_n \in \mathbb{P}^3$2 supported on spurious boundaries relates $Z_1, \dots, Z_n \in \mathbb{P}^3$3 and $Z_1, \dots, Z_n \in \mathbb{P}^3$4: $Z_1, \dots, Z_n \in \mathbb{P}^3$5. Dually, a similar boundary addition relates the duals of $Z_1, \dots, Z_n \in \mathbb{P}^3$6 and $Z_1, \dots, Z_n \in \mathbb{P}^3$7.

A plausible implication is that the Amplituhedron-Prime's structure can be generalized to higher $Z_1, \dots, Z_n \in \mathbb{P}^3$8 and loop orders, potentially elucidating the combinatorics and geometry of scattering amplitudes and their duals in planar $Z_1, \dots, Z_n \in \mathbb{P}^3$9 SYM(Herrmann et al., 2020).