Maximal Unitarity at Two Loops

TL;DR

The paper extends maximal generalized unitarity to two loops in massless four-point gauge-theory amplitudes, focusing on the planar double box. It develops a two-loop parametrization, solves the seven-propagator heptacut (plus the remaining degree of freedom) to obtain six on-shell solutions, and derives contour-constraint equations to ensure total-derivative and IBP reductions vanish appropriately. From these constraints, it derives compact master formulas for the coefficients c1 and c2 of the two double-box basis integrals in terms of residues of products of six tree amplitudes, with explicit validation in s- and t-channel channels for N=4,2,1 supersymmetry. The results agree with known leading ε^0 terms and establish a framework for extending the approach to D-dimensional cuts and full NNLO calculations.

Abstract

We show how to compute the coefficients of the double box basis integrals in a massless four-point amplitude in terms of tree amplitudes. We show how to choose suitable multidimensional contours for performing the required cuts, and derive consistency equations from the requirement that integrals of total derivatives vanish. Our formulae for the coefficients can be used either analytically or numerically.

Figures (10)

• Figure 1: The general quadruple cut in a one-loop box.
• Figure 2: The double box integral $P^{**}_{2,2}$.
• Figure 3: The heptacut double box.
• Figure 4: The six solutions to the heptacut equations for the two-loop planar double box.
• Figure 5: Schematic representation of contours for the coefficients of the two basis double boxes: (a) the scalar double box, $P^{**}_{2,2}[1]$ (b) the double box with an irreducible numerator insertion, $P^{**}_{2,2}[\ell_1\cdot k_4]$. The contours encircle the global poles distributed across the six kinematical solutions; the integers next to the contours indicate the winding number. Both representations are for the choice $u=\frac{1}{2}$ and $v=1$ in eqs. (\ref{['eq:heptacut_contours_diag_fam_1']}) and (\ref{['eq:heptacut_contours_diag_fam_2']}).