KLT orthogonality

KLT orthogonality states that vectors built from different solutions of the scattering equations are orthogonal under the field-theory KLT bilinear form. For generic massless kinematics, label the $(n-3)!$ solutions by $i$. Define the Parke-Taylor factor for a cyclic ordering $a_1,\dots ,a_n$ byLet $V_i(\alpha )$ evaluate this factor at solution $i$ in ordering $(1,\alpha ,n-1,n)$ and $U_j(\beta )$ in ordering $(1,\beta ,n,n-1)$. If $S[\alpha |\beta ]$ is the KLT momentum kernel, thenwith nonzero diagonal pairings generically.^[ref]

The pairing is bilinear, without complex conjugation; it is not an ordinary positive Hermitian inner product. At four points there is only one solution, so the off-diagonal statement is vacuous. At five points two solutions yield two vanishing cross-pairings. The proof relates the pairing to a generalized scattering-equation Jacobian whose rank drops from $n-3$ to $n-4$ for distinct solutions, forcing the relevant minor to vanish. This eliminates mixed-solution terms when combining gauge-theory amplitudes into gravitational amplitudes.^[ref]