Multi-Loop Positivity of the Planar ${\cal N}=4$ SYM Six-Point Amplitude

TL;DR

The work tests positivity for the six-point NMHV ratio in planar ${\mathcal{N}}=4$ SYM by leveraging Amplituhedron geometry and its positive kinematic regions, and then extends the test to the final, IR-finite amplitudes through analytic and extensive numerical checks up to five loops. It clarifies positive kinematic regions for MHV and NMHV amplitudes, expresses the NMHV ratio via the even function $V$, the parity-odd $\widetilde{V}$, and $R$-invariants, and studies the double-scaling limit to reveal monotonic, positive behavior. Across one-loop analytic results and multi-loop numerics, positivity holds in the Amplituhedron region, with evidence of monotonic radial growth away from collinear boundaries, while the MHV case remains positive under a BDS-like normalization but not for the standard remainder function. These findings indicate that positivity is preserved under momentum integration for these quantities and hint at a deeper geometric or contour-preserving mechanism that could enable a general positivity proof.

Abstract

We study the six-point NMHV ratio function in planar ${\cal N}=4$ SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a "radial" direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.

Figures (7)

• Figure 1: The coefficient functions $c_n^{(\ell)}(u,1)$ that multiply $\log^n(1/v)$ in the double-scaling limit at $\ell$ loops. Five loops is shown in blue, four loops in yellow, three loops in green, two loops in red, and one loop in purple.
• Figure 2: The coefficient functions $\tilde{c}^{(\ell)}_{0,k}(u)$ for the $w\rightarrow0$ edge of the double-scaling limit at $\ell$ loops. Five loops is shown in blue, four loops in yellow, three loops in green, two loops in red, and one loop in purple.
• Figure 3: The functions $c^{(\ell)}_{\ell-1}(u,u)$ and $c^{(\ell)}_{\ell-2}(u,u)$ governing the leading-log and next-to-leading-log behavior of the ratio function at $\ell$ loops in the double scaling limit. The variable $u$ has been shifted by $\tfrac{1}{2}$ to make it possible to plot on a log scale. Five loops is shown in blue, four loops in yellow, three loops in green, two loops in red, and one loop in purple.
• Figure 4: The three-loop coefficient functions $c_n^{(3)}(u,w)$ in the double-scaling limit, shifted to make it possible to plot them on a log scale. By plotting these functions against $\log u$ and $\log w$ we deform the $u+w=1$ line to the concave boundary seen in each plot.
• Figure 5: The leading-log coefficient functions $c_{\ell-1}^{(\ell)}(u,w)$ in the double-scaling limit from one to four loops, shifted to make it possible to plot them on a log scale. By plotting these functions against $\log u$ and $\log w$ we deform the $u+w=1$ line to the concave boundary seen in each plot.