Global Poles of the Two-Loop Six-Point N=4 SYM integrand

TL;DR

This work validates that the leading-singularity method in strictly four dimensions reproduces the full two-loop six-point MHV integrand of $\mathcal{N}=4$ SYM as predicted by recent loop-level recursion relations. By computing maximal and heptacuts of both general and factorized double-box topologies and solving the resulting linear systems, the parity-even and parity-odd components of the integrand are shown to match numerically with the recursion-based predictions across many random momenta. The analysis also clarifies the role of multidimensional contours and the overcomplete basis in determining integral coefficients, and discusses the practical utility of the intermediate results for constructing maximal-cut contours once complete IBP relations are known. Overall, the paper strengthens confidence in the consistency between modern four-dimensional recursion approaches and traditional unitarity-based methods for two-loop six-point amplitudes, while outlining avenues for future extensions to NMHV sectors and more extensive IBP-driven contour determinations.

Abstract

Recently, a recursion relation has been developed, generating the four-dimensional integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory for any number of loops and legs. In this paper, I provide a comparison of the prediction for the two-loop six-point maximally helicity-violating (MHV) integrand against the result obtained by use of the leading singularity method. The comparison is performed numerically for a large number of randomly selected momenta and in all cases finds agreement between the two results to high numerical accuracy.

Figures (21)

• Figure 1: The general double-box integral.
• Figure 2: The six kinematical solutions to the heptacut constraints for the double box topology in case I. For all solutions, the loop momentum parameters $(\alpha_1, \alpha_2, \beta_1, \beta_2)$ are set equal to the values given in eqs. (\ref{['eq:onshell-values-alpha-param1']})-(\ref{['eq:onshell-values-beta-param1']}). Any blob connecting more than three legs does not have a well-defined chirality and its sign should be ignored. For $\mathcal{S}_5$ and $\mathcal{S}_6$, the parameters $\beta_3$ and $\beta_4$ are determined by solving the on-shell constraint $(\ell_1 + \ell_2 + K_6)^2 = 0$ for the respective parameter. The on-shell values $P_1, Q_1$ etc. are functions of the external momenta; examples may be found in Sections \ref{['sec:heptacut_1']}, \ref{['sec:heptacut_2']}-\ref{['sec:heptacut_5']}, \ref{['sec:heptacut_8']}.
• Figure 3: The four kinematical solutions to the heptacut constraints for the double box topology in case II. For all solutions, the loop momentum parameters $(\alpha_1, \alpha_2, \beta_1, \beta_2)$ are set equal to the values given in eqs. (\ref{['eq:onshell-values-alpha-param1']})-(\ref{['eq:onshell-values-beta-param1']}). The on-shell values $P_1, Q_1$ etc. are functions of the external momenta and may be found in Section \ref{['sec:heptacut_6']}.
• Figure 4: The four kinematical solutions to the heptacut constraints for the double box topology in case III. In this case, because both $K_1$ and $K_2$ are massive, there are two solutions for $\gamma_1^\pm$ and therefore two pairs of flattened momenta $(K_{1\pm}^\flat, K_{2\pm}^\flat)$. For all solutions, the loop momentum parameters $(\alpha_1, \alpha_2, \beta_1, \beta_2)$ are set equal to the values given in eqs. (\ref{['eq:onshell-values-alpha-param1']})-(\ref{['eq:onshell-values-beta-param1']}). The on-shell values $P_1^\pm, Q_2^\pm$ etc. are functions of the external momenta and may be found in Section \ref{['sec:heptacut_7']}. The quantity $\xi^\pm$ is defined as
• Figure 5: The factorized double box integral with six massless external momenta.