= KLT orthogonality
{c}
<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 $a_1,\ldots,a_n$ by
$$
\operatorname{PT}(a_1,\ldots,a_n)=\frac1{\sigma_{a_1a_2}\sigma_{a_2a_3}\cdots\sigma_{a_na_1}},\qquad\sigma_{ab}=\sigma_a-\sigma_b.
$$
Let $V_i(\alpha)$ 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, then
$$
\sum_{\alpha,\beta}V_i(\alpha)S[\alpha|\beta]U_j(\beta)=0\qquad(i\ne j),
$$
with nonzero diagonal pairings generically.<Scattering Equations and KLT Orthogonality by Freddy Cachazo and others>[Section 5.2, equations (29)–(34), Proposition 2]{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.<Scattering Equations and KLT Orthogonality by Freddy Cachazo and others>[Section 5.1, Proposition 1; Sections 5.2 and 6.1]{ref}
Back to article page
