Two-Loop Maximal Unitarity with External Masses

TL;DR

This work extends the maximal generalized unitarity method to two-loop planar double-box integrals with up to three external massive legs. By formulating a rational spinor parametrization and solving heptacut equations, the authors identify eight global poles and derive constraint equations (Levi-Civita, IBP) to construct three class-(b) and two class-(c) master-contour projectors. These projectors yield unique coefficient formulas for all master integrals, demonstrated through detailed Higgs-operator and scalar-form-factor examples, and cross-validated against known two-loop results. The approach emphasizes internal-consistency with total-derivative vanishing and provides a practical, analytic/numeric pathway for extracting two-loop double-box coefficients in gauge theories with external masses.

Abstract

We extend the maximal unitarity method at two loops to double-box basis integrals with up to three external massive legs. We use consistency equations based on the requirement that integrals of total derivatives vanish. We obtain unique formulae for the coefficients of the master double-box integrals. These formulae can be used either analytically or numerically.

Figures (7)

• Figure 1: The double-box integral $P^{**}_{2,2}$.
• Figure 2: The short-side two-mass double box. Single lines indicate massless legs, and doubled lines massive legs (here the two legs on the left-hand side).
• Figure 3: Other two-mass double-boxes: (i) the diagonal one (ii) the long-side one. Single lines indicate massless legs, and doubled lines massive legs (in (i), the lower-left and upper-right legs, and in (ii), the two lower legs).
• Figure 4: A representation of the solution space for the class (b) heptacut equations, showing the four independent solutions ${\cal S}_i$, and the locations of the eight global poles ${\cal G}_j$. The small white, black and gray blobs indicate the pattern of chiral, antichiral and nonchiral kinematics, respectively, at the vertices of a double-box heptacut.
• Figure 5: A representation of the solution space for the class (c) heptacut equations, showing the six independent solutions ${\cal S}_i$, and the locations of the eight global poles ${\cal G}_j$. The small white and black blobs indicate the pattern of chiral and antichiral kinematics, respectively, at the vertices of a double-box heptacut.