[Paper Review] Positive Geometries for One-Loop Chiral Octagons

Enrico Herrmann, Cameron Langer+2 more|arXiv (Cornell University)|Jul 23, 2020

Axial and Atropisomeric Chirality Synthesis4 citations

TL;DR

This paper introduces a new class of positive geometries for one-loop amplitudes in planar $χ=4$ super Yang-Mills theory, showing that maximal sign-flip regions correspond to chiral octagon integrands—key building blocks for infrared-finite, dual-conformal-invariant amplitudes. The work establishes the first direct link between positive geometry and Feynman integrals, revealing that logarithmic forms on these geometries yield chiral one-loop integrals with $d\log$ representations.

ABSTRACT

Inspired by the topological sign-flip definition of the Amplituhedron, we introduce similar, but distinct, positive geometries relevant for one-loop scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. The simplest geometries are those with the maximal number of sign flips, and turn out to be associated with chiral octagons previously studied in the context of infrared (IR) finite, pure and dual conformal invariant local integrals. Our result bridges two different themes of the modern amplitudes program: positive geometry and Feynman integrals.

Motivation & Objective

• To extend the positive geometry framework beyond tree-level amplitudes to one-loop scattering amplitudes in planar $χ=4$ SYM.
• To identify and classify positive geometries associated with chiral octagon integrands, which are known to yield infrared-finite, dual-conformal-invariant amplitudes.
• To establish a direct correspondence between logarithmic differential forms on these geometries and chiral one-loop integrals.
• To demonstrate that the maximal sign-flip regions in the one-loop Amplituhedron construction correspond precisely to chiral octagons, unifying two themes in the amplitudes program.

Proposed method

• Adapts the topological sign-flip definition of the Amplituhedron to one-loop kinematics using momentum-twistor variables and $\textrm{SL}(4)$ invariants.
• Defines one-loop positive geometries via sign-flip conditions on sequences of four-brackets $\langle abcd\rangle$, with maximal sign flips corresponding to chiral octagons.
• Constructs logarithmic differential forms $\Omega^{(\mathcal{S})}$ as products of $Z$-forms (on momentum twistors) and $(AB)$-forms (on loop momenta), ensuring $d\log$ structure.
• Uses $d\log$ representations to triangulate the geometry and show cancellation of spurious poles, confirming consistency of the form.
• Applies the framework to MHV and non-MHV helicity configurations, showing that $Z$-forms become nontrivial for $k>0$.
• Demonstrates that chiral octagon integrands arise naturally as the logarithmic forms on these maximal sign-flip geometries, with explicit expressions for $\omega_Z$ and $\omega_{AB}$.

Experimental results

Research questions

• RQ1 Which one-loop positive geometries in planar $χ=4$ SYM correspond to chiral octagon integrands?
• RQ2 How do sign-flip conditions in the Amplituhedron framework constrain the structure of one-loop amplitudes?
• RQ3 What is the role of $Z$-forms and $(AB)$-forms in constructing logarithmic differential forms for chiral integrals?
• RQ4 Why do maximal sign-flip regions stabilize at eight points, and what does this imply for the complexity of one-loop amplitudes?
• RQ5 Can the positive geometry framework generate new classes of infrared-finite $d\log$ integrands beyond one loop?

Key findings

• The maximal sign-flip regions in the one-loop Amplituhedron correspond exactly to chiral octagon integrands, which are known to be infrared-finite and dual-conformal invariant.
• The logarithmic differential form $\Omega^{(\mathcal{S})}$ on these geometries is given by a product of $Z$-forms and $(AB)$-forms, explicitly constructed for chiral octagons.
• Spurious poles such as $\langle 1235\rangle$, $\langle 1356\rangle$, and $\langle 1345\rangle$ cancel nontrivially between numerator and denominator terms, confirming the consistency of the $d\log$ structure.
• For MHV amplitudes, the $Z$-form is trivial ($\omega_Z = 1$), but for non-MHV configurations, nontrivial $Z$-forms emerge, indicating richer geometric structure.
• The framework reveals that chiral octagon integrals are naturally associated with local positive geometries distinct from the full Amplituhedron, with fixed geometry size even as external particle count increases.
• The result provides the first direct connection between positive geometry and Feynman integrals, opening a path to generating $d\log$ integrands for higher-loop amplitudes.