The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

TL;DR

The paper derives an analytic expression for the rescaled one-loop six-dimensional hexagon integral $\tilde{\Phi}_6$ in terms of classical polylogarithms and links it to integrals $\Omega^{(1)}$ and $\Omega^{(2)}$ appearing in the six-point MHV amplitudes of planar $\mathcal{N}=4$ SYM. A set of first-order differential equations connects $\Omega^{(2)}$, $\tilde{\Phi}_6$, and $\Omega^{(1)}$, with $\tilde{\Phi}_6$ serving as an intermediate object between the two, enabling a full determination of the three-variable function $\Phi_6(u_1,u_2,u_3)$. The authors provide an explicit formula for $\Phi_6$ in terms of a compact combination of degree-3 functions $L_3$ and $J$, defined via dilogarithms and their shifted variants, and verify the differential equations both analytically and via symbol calculus. The work illuminates structural connections between one- and two-loop amplitudes, suggests extensions to higher transcendentality and massive kinematics, and points to Wilson-loop representations as a promising avenue for further insight into $\mathcal{N}=4$ SYM amplitudes.

Abstract

We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral $\tildeΦ_6$ with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$ super-Yang-Mills theory, $Ω^{(1)}$ and $Ω^{(2)}$. The derivative of $Ω^{(2)}$ with respect to one of the conformal invariants yields $\tildeΦ_6$, while another first-order differential operator applied to $\tildeΦ_6$ yields $Ω^{(1)}$. We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$ super-Yang-Mills.

Figures (2)

• Figure 1: Three dual conformal integrals which are related to each other by the action of first-order differential operators, as discussed in the text. The labels $i,j,1,2,\ldots,6$ are indices $k$ for dual (or region) coordinates $x_k$. Solid lines indicate propagators; dashed lines indicate numerator factors of $x_{ai}^2$ or $x_{bi}^2$, as explained in the text. The central integral $\tilde{\Phi}_6$ has no such numerator factors, but is evaluated in dimension $D=6$ instead of $D=4$. The standard hexagon integral $H$ is rescaled to obtain a dual conformal invariant integral $\Phi_6$, which is rescaled once again to obtain the pure degree 3 function $\tilde{\Phi}_6$.
• Figure 2: Interpretation of the hexagon integral as a line integral, according to eqs. (\ref{['hex-wl-1']}) and (\ref{['hex-wl-2']}).