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Ellis Ye Yuan co-developed the scattering equations with Freddy Cachazo and Song He, connecting massless-particle kinematics with marked points on a sphere.^[ref] In a common convention,with conserved massless momenta $k_a$ and puncture coordinates ${\sigma }_a$. The equations provide algebraic structure for scattering calculations rather than trajectories of particles moving on that auxiliary sphere.

Freddy Cachazo co-developed the scattering equations with Song He and Ellis Ye Yuan. Their work connected kinematic invariants with punctures on a sphere and provided an inductive solution method, together with a proof of KLT orthogonality.^[ref]

The resulting auxiliary geometry organizes calculations of massless-particle scattering without treating that sphere as physical spacetime.

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]

In the CHY formalism, the scattering equations relate massless external four-momenta $p_i$ to distinct marked points $z_i$ on the Riemann sphere:All momenta are taken outgoing; incoming particles have the opposite signed four-momentum. The dot product uses the spacetime metric. ^[ref]

Pairing terms with interchanged indices and using momentum conservation givesFor generic kinematic data, only $n-3$ equations are independent. Under a Möbius transformation $z\mapsto (az+b)/(cz+d)$ with $ad-bc=1$, nullness and momentum conservation give $E_i\mapsto (c{z}_i+d{)}^2{E}_i$. Thus three marked points can be fixed by this projective symmetry.

The equations also arise by making the Koba-Nielsen factor's logarithm stationary in the massless high-energy limit. The massless conditions are essential to the displayed projective-covariance argument; massive generalizations require a modified system.

Song He, with Freddy Cachazo and Ellis Ye Yuan, established KLT orthogonality for objects built from solutions of the scattering equations. The orthogonality uses a specified bilinear form, not an arbitrary Euclidean inner product.^[ref]

This relates alternative organizations of particle-scattering data while keeping the momentum constraints explicit.